Imagine_Maths_CB_Grade8_Sales_Sample_Part1 by Uolo

Rational Numbers

Let's Recall In our previous grades, we learned about natural numbers and whole numbers. We know that integers are positive and negative whole numbers. We also know how to read, write and represent fractions and decimals. We know that these are all rational numbers. Let us recap how to add and subtract integers, fractions and decimals. Adding Integers

Subtracting Integers

Like Sign

Unlike Sign

Add and keep the sign.

Subtract and keep the larger number sign.

4+2=6

Change the sign of subtrahend and solve. –3 – (–2) = –3 + 2 = –1

(–4) + 3 = –1

(–4) + (–2) = –6 Adding and Subtracting Decimals • Convert to like decimals • Align the decimal points • Add or subtract 1 2 .6 3 + 1 .3 0 1 3 .9 3

1 2 .6 3 − 1 .3 0 1 1 .3 3

Adding and Subtracting Fractions Like Fraction

Unlike fraction

• Add or subtract the numerators.

• Convert to like fraction by taking the LCM.

• Keep the common denominator.

• Add or subtract as done in like fractions.

2 4 2+4 6 + = = 5 5 5 5

2 2 10 − 6 4 – = = 3 5 15 15

Letʼs Warm-up Fill in the blanks. 5 + 2 4 3

__________________

2 52.31 + 61.2

__________________

3 − 23 + 69

__________________

7 − 5 8 4

__________________

5 89 − 47.25

__________________

I scored _________ out of 5.

Understanding Rational Numbers Shubhi and Rishabh collected data on the temperature of different cities across the world in December. Rishabh: Shubhi, can you convert the temperature of Yakutsk and Dubai in fractions? Shubhi: Let me try Rishabh! We can write −25.5°C as −255 °C and 22.5°C as 255°C. 10 10 Rishabh: But −255°C is not a fraction! 10 Both the kids got confused about what should they call this number. Let’s help them out!

Temperature in ℃

Real Life Connect

22.5

20 10 0

–10

–20

–30

–25.5 Yakutsk

Dubai

–18.3

–14.2

Fraser

Golmud

City

Rational Numbers On converting − 25.5°C and 22.5°C to fractions, Shubhi got −255°C 10 and 255°C. These numbers are called rational numbers. 10 A rational number is a number that is expressed as p, where p and q q are integers and the denominator should never be equal to zero.

Rational

Any number that can be written as function –5.25 56 12

For example, 12.5, − 15, 25, 6 and −8 are all rational numbers. 7 9

Whole

–12 10 7.5

All positive integers and zero 0, 1, 2, 3, 4,....

Natural

Identifying and Representing Rational Numbers

All positive integers, excluding zero 1, 2, 3, 4,....

We know how to represent integers, fractions and decimals on a number line. Let us see how to represent rational numbers on a number line. We have marked the number line in equal intervals of 0.5 keeping 0 in the center.

Integers

All positive and negative whole numbers, and zero ....-3, –2, –1, 0, 1, 2, 3,....

Rational Number −1 = − 0.5 2 −1 2

−2 −1.5 −1 −0.5

Integer

Example 1

0.5

1.5

2.5

Decimal Number

Represent 5 and − 4 on the number line. 3 3 We can write 5 as 1 1. So, it will lie between 1 and 2 on the number line. Divide the space between 1 and 2 3 3 into 3 equal parts and mark 5. 3

We can write −4 as − 11. So, it will lie between –1 and –2 on the number line. Divide the space between –1 3 3 and –2 into 3 equal parts and mark –4. 3 −5 3

Example 2

−4 3

−3 −2 3

−1

Mark − 2, 1, 1.75 and –2.25 on the number line.

−2

−1

The number line has been marked at equal intervals of 1 in both directions. 4 4 7 − 8 − 9 −2= , 1 = , 1.75 = and − 2.25 = 4 4 4 4

−10 −9 4 4 Do It Together

Represent

−8 4

−7 4

−6 4

−5 4

−1 −3 −2 4 4

−1 4

1 4

2 4

1 8

2 8

3 4

4 4

5 4

6 4

7 4

9 4

10 4

−2 5 , , −1 and 0.5 on the number line. 8 8

–7 8

–6 8

–4 8

–1 8

4 8

Comparing and Ordering Rational Numbers Rational numbers can be compared by changing them to the same form. Let us recall how to write rational numbers in standard form and equivalent rational numbers. Standard Form of Rational Numbers A rational number is said to be in standard form if: • both the numerator and denominator contain no common factors other than 1. • the denominator has a positive integer. To convert rational numbers to standard form: • divide both numerator and denominator with their HCF. For example, the standard form of 6 = 6 ÷ 3 = 2 . 9 9÷3 3 Equivalent Rational Numbers

Think and Tell

Is pi a rational number?

Two rational numbers are equivalent if their standard forms are equal. To find the equivalent rational number of a rational number, we multiply or divide the numerator and denominator by the same natural number. 1 1×2 2 1×3 3 2 3 1 For example, = = ; = . So, and are equivalent to . 4 4 × 2 8 4 × 3 12 8 12 4 Two or more rational numbers can be compared and ordered using the steps given.

Step 1: Write all the rational numbers in the same form. Chapter 1 • Rational Numbers

p q

Step 2: If the rational numbers are in form and their denominators are different, find the LCM of the denominators. Step 3: Find equivalent rational numbers with denominator as the LCM.

Step 4: C ompare the numerators of the equivalent rational numbers. The rational number with the greater numerator is the greater rational number.

2 5 and using the above steps. 3 7 Both the rational numbers are already in the same form, so we will find the LCM of 3 and 7 = 21. 2 2 × 7 14 5 5 × 3 15 = = and = = 3 3 × 7 21 7 7 × 3 21 14 15 As, 14 < 15. So, < . 21 21 2 5 Hence, < . 3 7 Now what if the rational numbers are in different forms? How do we compare a fraction with a decimal or a decimal with an integer? Let us compare

To compare different kinds of rational numbers, we convert them into the same form. −7 . 3 p −7 The comparison can be done by either converting − 2.5 to form or by converting to integers. q 3 Let us compare − 2.5 and

Method 1

Method 2

−25 −5 = 10 2 LCM of 3 and 2 = 6

−7 = −2.33 3 As, − 2.5 < − 2.33

− 2.5 =

−5 −5 × 3 −15 −7 −7 × 2 −14 . = = ; = = 2 2×3 6 3 3×2 6 −15 −14 As, −15 < −14. So, < 6 6 −7 Hence, −2.5 < . 3

Let us now arrange the rational numbers LCM of 7, 3, 9 and 5 = 315

So, −2.5 <

−7 . 3

−5 7 3 , , and in ascending order. 7 3 5

−5 × 45 −225 7 × 105 735 2 × 35 70 3 × 63 189 = , = , = and = . 3 × 45 315 3 × 105 315 9 × 35 315 5 × 63 315 Compare the numerators of the rational numbers. −225 < 70 < 189 < 735 The rational numbers can be written as

−225 70 189 735 < < < . 315 315 315 315 −5 2 3 7 So, the rational numbers in ascending order are < < < . 7 9 5 3 −4 Give any three equivalent rational numbers to . 5 We can write the rational numbers in ascending order as

Example 3

−4 −4 × 2 −8 = = 5 5 × 2 10 So, the three equivalent rational numbers of

−4 − 4 × 3 −12 = = 5 5×3 15 −4 −8 −12 −16 are , , and . 5 10 15 20

−4 −4 × 4 −16 = = 5 5×4 20

Example 4

4 Suman and Kamal bought equal amounts of fruit in their lunch box. Suman ate of the fruit while 9 8 Kamal ate of the fruit. Who ate more fruit? 11 To compare the given numbers, we will find the LCM of 9 and 11. LCM (9, 11) = 99 Write the equivalent rational numbers. Compare the numerators 44 < 72

4 × 11 44 8 × 9 72 = and = . 9 × 11 99 11 × 9 99

Think and Tell How is

4 8 < . 9 11 Therefore, Kamal ate more fruit than Suman.

−4 different from 4 ? 5 −5

Hence,

Do It Together

13 8 , , 1.7 in descending order. 6 3 p To compare and order the rational numbers, first convert them in form. q Therefore, the rational numbers can be written as: _____________________ Arrange the rational numbers − 3,

Find the LCM of the denominators and write the equivalent rational numbers. The LCM of 1, 6, 3 and 10 is _____________________. The equivalent rational numbers are: ____, ____, ____, ____ Compare the numerators. _____________________ Descending order = ____>____>____>____

Rational Numbers Between Two Rational Numbers Rational numbers between two rational numbers can be found using their mean or equivalent rational numbers. By Finding the Mean

By Finding Equivalent Rational Numbers

p m p m If q and n are two rational numbers and q < n , 1 p m  p m  +  is a rational number between q and n . 2  q n 

For example, a rational number between −21 +10   −3 2 −11 and = 1  −3 + 2  = 1 =   5 7 2 5 7 2 35 70

(

Example 5

)

• Find the LCM of the denominators and write the equivalent rational numbers.

• Write the rational numbers in increasing order. For example, to find the rational numbers between

1 and 8, we find the LCM of 3 and 9 and write the 3 9 1 3 8 8 equivalent rational numbers as = and = . 3 9 9 9 3 8 H ence, the rational numbers between and are 9 9 4, 5, 6 and 7 . 9 9 9 9

Find two rational numbers between 2 and 3. The rational number between 2 and 3: =

Chapter 1 • Rational Numbers

1 (2 + 3)= 2

5 5 ;2 < < 3 2 2

The rational number between 2 and 5: = ;2 < <  2 + =  = 2  2  2  2  4 4 2 2 1

5

19

Hence, two rational numbers between 2 and 3 are 5 and 9. 2 4 Give four rational numbers between 1 and 1 . 3 2 LCM of 3 and 2 = 6.

Example 6

Hence,

1 2 1 3 = and = 3 6 2 6

Here, the new equivalent rational numbers do not have any numbers between them. So, we multiply and divide each of them by 5. 2 2 × 5 10 3 3 × 5 15 = = and = = 3 6 × 5 30 6 6 × 5 30

Four rational numbers between Do It Together

10 15 11 12 13 14 and = , , and . 30 30 30 30 30 30

7 Give five rational numbers between − 4 and . 12 9 LCM of 9 and 12 = ____, hence ____ − 4 = and 7 = ____ 9 12 Five rational numbers between − 4 and 7 = ________________________________ 9 12

Do It Yourself 1A 1 Represent the rational numbers on the number line. a

4 5

b -

2 4

5 3

d -

6 5

2 Which rational number does point Q represent on the number line? Q

-2

3 Convert the rational numbers into their standard form. a

3 9

-2 16

-1 c

8 -28

-16 -42

15 -21

4 Give three equivalent rational numbers for the rational numbers. a

5 8

5 Compare and fill in the blanks.

-9 12

6 8

1 3 ______ 4 5

-5 7 ______ 8 12

c - 0.4 ______

-3 14

2 16 ______ 9 72

-6 -5 ______ 11 12

14 ______ 2.8 5

g -3.2 ______

18 5

53 19 ______ 7 2

6 Arrange the rational numbers in ascending order. 9 1 8 5 a -3.5, , 2 , , 5 4 7 2

-3 2 7 , 1.5, 4.2, 2 , 2 5 5

5 1 , -0.75, 1.25, 1 , 7 9 6 10

8 2 , -2, 1.25, 1 , 13 9 4 4

16 1 , 3 , -2.5, 3, -18 3 4 5

3 -25 36 17 , 2.1, , , 5 2 5 2

7 Circle the rational numbers that lie between -3 and -2. -13 5

-16 8

-21 6

-18 8

-14 -26 and . 9 12 1 Five times the reciprocal of a number plus is 8. Find the number. 2

8 Give four rational numbers between 9

Word Problem 1

7 5 2 m of red cloth and m of green cloth. She used of the total cloths. Which 8 6 5 property can be used to find the amount of cloth used? Use the property and verify your answer. Simran bought

Operations on Rational Numbers Real Life Connect

Shubhi and Rishabh now started looking for the temperature of their city over the past 5 days. They recorded the data in the form of rational numbers.

Day

Temperature

Monday

-3 °C 2 16 °C 5 9 °C 5 23 °C 5 -3 °C 4

Both of them performed various operations on the rational numbers and found different data. Let us see how!

Tuesday

Addition and Subtraction of Rational Numbers

Thursday

Addition of Rational Numbers

Friday

Wednesday

The addition of rational numbers is similar to the addition of fractions. However, we first have to convert all the rational numbers to be added into rational numbers with positive denominators. Let us use the data collected by Rishabh and Shubhi and look at the cases and steps of addition of rational numbers. Same Denominators

16 9 + 5 5 (16 + 9) 25 = = =5 5 5

Different Denominators

2 −3 + 9

2 5 LCM of 2 and 5 is 10.

Write the equivalent rational number with the denominator as LCM. −3 × 5 −15 9 × 2 18 = and = 10 2×5 5 × 2 10 −15 18 −15 + 18 3 + = = 10 10 10 10

Chapter 1 • Rational Numbers

Adding Rational Numbers with Negative Denominators Let us add 16 and 3 5 −4

Converting 3 into a rational number with a positive denominator, we get 3 × −1 = −3 −4 −4 −1 4 LCM of 5 and 4 = 20 Write the rational numbers with denominators as 20. 16 × 4 = 64 and −3 × 5 = −15 5 × 4 20 4×5 20 So, 16 + (−3) = 64 + −15 = 64 − 15 = 49. 5 4 20 20 20 20 Example 7

Add

18 −19 , and 2.5. 4 8

2.5 can also be written as 25. 10 LCM of 4, 8 and 10 is 40.

Write the rational numbers with the denominator as 40. 18 × 10 = 180, −19 × 5 = −95 and 25 × 4 = 100 4 × 10 40 8 × 5 40 10 × 4 40 Hence, 18 + −19 + 25 4 8 10 180 + −95 + 100 = 180 − 95 + 100 = 185 = 37 40 40 40 40 40 8 Example 8

A shopkeeper sold 2

1 3 2 kg, 5 kg 1.5 kg and 4 kg of rice to four customers. Find the total amount of 4 8 3

rice sold by the shopkeeper.

The weights can be expressed in rational numbers as: 2

1 9 3 43 15 3 2 14 = , 5 = , 1.5 = = and 4 = 4 4 8 8 10 2 3 3

Total weight of rice sold =

9 43 3 14 + + + 4 8 2 3

LCM of 4, 8, 2 and 3 = 24 54 129 36 112 54 + 129 + 36 + 112 331 Hence total rice sold = + + + = = kg 24 24 24 24 24 24 Do It Together

1 Add 15, 23 and 3.6. 4 5

2 Add 5, 6 and ‒ 2.8. 3 ‒13

Subtraction of Rational Numbers We know that subtraction is the opposite or inverse of addition. To subtract two rational numbers, we simply add the additive inverse of the rational number to be subtracted to the first rational number. Also, the rational numbers should have a positive denominator. 17 7 17 -7 34 + (-21) 34 - 21 13 For example - = + = = = 6 6 3 2 3 2 6

What if the rational numbers are in different forms? We will first convert the numbers into the same form. 23 Let us subtract from 7.85 5 Method 1

Convert

23 = 4.6 5

Method 2

Convert 7.85 to a fraction.

23 to decimal. 5

7.85 =

785 100

785 23 785 - 23 = + 100 5 100 5

7.85 - 4.6 = 3.25

785 + (- 460) 785 - 460 325 13 = = = 100 100 100 4

What if we have more than two rational numbers with different signs between them? Let us find out! Simplify

2 5 1 4 + - + 3 9 6 3

To simplify the above statement, we will find the LCM of all the denominators, and convert them in like rational numbers. LCM of (3, 9, 6) = 18 Hence,

2 5 - 1 4 12 + 10 - 3 + 24 43 + + + = = 3 9 6 3 18 18

Simplify

Example 9

7 8 -3 -9 + + 6 9 12 15

Example 10

7 8 -3 -9 7 8 -3 + + = + + additive inverse of 6 9 12 15 6 9 12 7 8 3 -9 ⇒ + + + (LCM of 6, 9, 12 and 6 9 12 15 15 is 180.)

Do It Together

⇒

210 160 45 - 108 + + + 180 180 180 180

⇒

210 + 160 + 45 - 108 307 = 180 180

1 Subtract 12 from 6.25. 5

Chapter 1 • Rational Numbers

521 kg. Suppose 12 the empty drum weighs 12.5 kg. What is the weight of the sugar in the drum? 521 Weight of drum with sugar = kg 12 125 25 Weight of empty drum = 12.5 kg = = kg 10 2 A drum full of sugar weighs

Weight of sugar =

521 25 521 - 25 - = + 12 2 12 2

521 - 150 + (LCM of 12 and 2 is 12.) 12 2

521 - 150 371 = kg 12 12

2 Subtract from 5 from 31. ‒2 2

Properties of Addition and Subtraction of Rational Numbers Closure Property If

For example,

a c a c a c and are rational numbers, then + and - are also rational numbers. b d b d b d

-5 3 -7 -1 + = or (Rational number) 7 14 14 2

Commutative Property

For Addition If

For example, - 4.5 -

a a c c a c and are rational numbers, then + = + . b b d d b d For example,

1 = -5 (Rational number) 2

For Subtraction

a c a c c a and are rational numbers, then - ≠ - . b d b d d b

5 3 13 3 5 13 + = and + = 7 14 14 14 7 14

For example,

8 7 1 7 8 -1 - = but - = 9 9 9 9 9 9

Associative Property

For Addition

For Subtraction

a c e a c e If , and are rational numbers, then + + b d f b d f a c e = + + . b d f For example,

a c e a c e , and are rational numbers, then - b d f b d f a c e ≠ - - . b d f

5 5 32 2 3 7 + + = + = 2 5 10 10 2 10

For example,

2 3 5 2 28 32 + + = + = 5 10 2 5 10 10 Additive Identity If

For example, Example 11

a a is a rational number, then + b b -a = 0. b For example,

Verify if m - n is a rational number if m = m-n=

Example 12

-2 -2 +0= 3 3

4 3 5 4 -1 9 - - = = 5 5 10 5 10 10

Additive Inverse

a a is a rational number, then + 0 b b a a =0+ = . b b

5 4 3 1 5 –3 – = = 10 5 5 5 10 10

5 -5 + =0 8 8

Subtraction Property of Zero If

a is a rational number, then b a a -0= . b b For example,

–9 6 and n = . 11 22

-9 6 -18 - 6 -24 -12 = = = = rational number 11 22 22 22 11

5 3 7 and are three rational numbers, then prove that they are not associative under subtraction. 4 8 12

If ,

a c e a c e - ≠ b d f b d f 5 3 7 7 7 7 5 3 7 5 - 5 35 So, - = = and - = = 4 8 12 8 12 24 4 8 12 4 24 24 Associative property for subtraction states that

7 35 ≠ ; hence rational numbers are not associative under subtraction. 24 24

Do It Together

Fill in the blanks using the properties of rational numbers. 1 5 - _______ = 5 2 2

3 3 -0= 5 5

2 5 + -3 = -3 + _______ 2 9 9

3 12 + _______ = 0 25

Do It Yourself 1B 1 Solve. a

4 6 + 7 7

d -2 + 4 + 2.5

8 -2 11 11

9 2 - 1.3 7 -9

e 8 + (-24) + 14.5

2 Solve. a

5 -7 + 9 11

b f

9 38 13 39

3 What decimal should be added to

f 17 + 18.44 + (-12)

8 -14 7 15 11 5

-12 -13 + 2 15

-5 7 + 12 15

13 9 - 2.1 -4 16

-13 -6 8 15

17 - 16 2 5 3 7

21 -27 to get ? 5 6

4 What fraction should be subtracted from 8.5 to get (-12.25)? 3 7

2 3

5 A rope is 5 m long. Another rope is 1 m shorter than it. What is the total length of the ropes? 6 Ravi earned ₹480 in 1 day. He spent ₹ he save?

153 253 on tea and snacks, ₹ on food and the rest he saved. How much did 2 2

7 Name the property illustrated through each of the operations. a

a c - = a rational number b d

-12 7

a c e a c e + + = + + b d f b d f

a a -0= b b

13 2

5 21 d -9 14

8 Find the additive inverse of the rational numbers. a

9 Simplify

5 -3 6 -3 2 8 + - + - 6 8 7 14 5 -15

8 9

5 3

7 6

10 Find the rational number that must be added and subtracted so that they will make the sum + - to the nearest whole number.

11 Era has the given weight of grains in her pantry.

Wheat 25.5 kg

a How much wheat and rice does Era have?

Rice 12

2 kg 3

Corn 37 kg 2

b What is the difference between the weight of wheat and corn?

Word Problems 1

A carpenter bought a piece of wood 12

Jimmie bought 18 kg of flour. He used 3

How long is the piece of wood now?

2 5 feet long. He then cut off feet from both ends. 3 8

1 2 7 kg in the first week, 4 kg in the second week, 4 kg 5 3 8 in the third week and the rest in the last week. How much flour did he use in the last week?

Chapter 1 • Rational Numbers

3 4

15 kg of apples 2.6 kg of bananas and some oranges from the market. 4 Suppose she purchased 9 kg of fruit. What is the weight of the oranges purchased? 2 A school prepared 25.5 litres of juice for its players. The players drank 8 litres in the first 5 3 half. In the second half, the players drank 2 litres more than in the first half. What is the 8 volume of the juice left at the end? Suhani purchased

Multiplication and Division of Rational Numbers Multiplication of Rational Numbers The multiplication of rational numbers is similar to that of fractions. Product of numerators Product of two rational numbers = Product of denominators For example,

12 5 60 7 -3 -21 -7 × = and × = = 13 7 91 9 8 72 24

What if the rational numbers are in different forms? -4 Let us multiply and 0.56 13 56 14 We will first convert them into a similar form as 0.56 = = . 100 25 (–4) 14 (–56) × = 13 25 325 Example 13

Example 14

13 and -8.5. 25 p -85 Write the numbers in the form ; hence, –8.5 = . q 10 13 -85 -1105 × = 25 10 250 -1105 -221 HCF of 1105 and 250 = 5; hence = 250 50 145 231 The length and width of a rectangular park are m and m. 3 9 What is the perimeter of the park? Find the product of

Perimeter of rectangle = 2 × (L + B) = 2 2×

145 231 + 3 9

435 + 231 2 666 1332 = × = = 148 m 9 1 9 9

Hence, the perimeter of the rectangular park is 148 m. Do It Together

1 Multiply 5 and 16 9 -7

2 Multiply 2.5 and -19 -3

Remember! The product of two rational numbers with similar signs is positive and with opposite signs is negative.

Error Alert! p Convert the numbers into q form and then multiply. 2 8.1 × 2.7 = 3 3

2 27 54 18 × = = 3 10 30 10

Division of Rational Numbers We know that division is the opposite of multiplication. To divide a rational number by another rational number, we multiply the dividend by the multiplicative inverse of the divisor. That is

a c a d ÷ = × b d b c Divisor

Dividend For example,

Did You Know?

15 3 15 6 90 15 ÷ = × = = 4 6 4 3 12 2

The size of Earth is

To divide rational numbers into different forms we first convert them to the same form and then divide them. For example,

13 13 2 13 1 13 ÷ 2 = ÷ = × = 5 5 1 5 2 10

8 951 m of cloth for ₹ What is 3 4 the cost of the cloth per meter? Meeta bought

Example 15

Cost of

Example 16

12 13 -5 3 9 × ÷ × ÷ 5 18 8 5 8

951 8 951 3 2853 ÷ = × = m. 4 3 4 8 32

1 Divide: –15 ÷ 21 8 5

Simplify

Solve the numbers inside the brackets first,

8 951 m of cloth = ₹ 3 4

Cost of 1 m of cloth =

Do It Together

the size of Jupiter.

1 times 11

12 13 -8 3 8 × × × × 5 18 5 5 9

12 -104 24 -29952 -1664 × × = = 5 90 45 20250 1125

2 Divide: 3.5 ÷

-19 25

Properties of Multiplication and Division of Rational Numbers Closure Property For Multiplication

For Division

a a c c and are rational numbers, then × will also be b d b d a rational number. If

For example,

−6 −3 2 × = (rational number) 5 2 10

a a c c and are rational numbers, then ÷ is not b d b d necessarily a rational number. If

For example,

6 ÷ 0 = not defined (not a rational number) 11

Commutative Property For Multiplication If

a c c a a c and are rational numbers, then × = × . b d d b b d

For example, 1 × 2 = 2 and 2 × 1 = 2 6 7 6 42 7 42 Chapter 1 • Rational Numbers

For Division If

a c c a a c and are rational numbers, then ÷ ≠ ÷ . b d b d d b

For example,

5 2 25 20 2 5 = = ÷ ÷ but 10 5 5 10 20 25

Associative Property For Multiplication If

a, c

and

For Division

are rational numbers, then

b For example,

2 2

∴

d 5 5

14 1 14 × = 10 2 20

2 14 7 × = 2 10 20

2 1 7 × × 2 2 5

a, c

and

are rational numbers, then

≠

For example, 3 ÷ 2 ÷ 1 = 21 ÷ 1 = 21 8 4 7 8 8 3

∴

8 ÷

16 21 3 ÷ = 4 7 96

≠

2 1 3 ÷ ÷ 4 7 8

Distributive Property Multiplication over addition and subtraction a, c

and

are rational numbers, then a

a, c

a e a c × + × b f b d and a c a e e a c × − × − × = b d f b f b d b

Division over addition and subtraction

and

× 1 = 1 × 3

×1=

÷1=

c e a e ÷ + ÷ d f b f and c e e a e ÷ = ÷ − ÷ d f f b f

2 −1 2 −4 = ÷ = 4 4 4 8 and 1 2 3 2 4 12 −4 ÷ ÷ = = 2 4 4 8 4 4 8 1 3 2 4

Multiplicative Inverse

a is a rational number, then b b

2 5 2 20 = ÷ = 4 4 4 8 and 1 2 3 2 4 12 20 ÷ + ÷ = + = 2 4 4 8 4 4 8

× 0 = 0 ×

a is a rational number, then b

= 0.

6 ×0=0 11

If ÷ (−1) =

= 1. =1

Division by itself and inverse

a is a rational number, then b

−

For example, 1 + 3 2 4

Division by 1 and −1 If

Multiplicative Property of 0

a is a rational number, then b b

1 3 2 1 1 1 × = × = 2 2 4 4 8 4 and 1 3 1 2 3 2 1 × × = - = 2 4 8 8 8 2 4 Multiplicative Identity

For example, 1 × 3 + 2 = 1 × 5 = 5 2 2 4 4 8 4 and 1 3 1 2 3 2 5 × + × = + = 2 4 8 8 8 2 4

are rational numbers, then

−a b

a is a non-zero rational number, then b

−a b

= −1

Example 17

6÷ 8 – 7 ÷ 8 using the distributive property of division over subtraction. 5 11 12 11 a c a c e e e − − ÷ ÷ = The distributive property of division over subtraction states that ÷ b d b d f f f So, 6 ÷ 8 − 7 ÷ 8 = 6 − 7 ÷ 8 5 11 11 12 12 5 11 Solve

8 = 11

⇒ 6 × 12 − 7 × 5 ÷ 60

⇒ Example 18

8 37 37 11 407 ÷ × = = 60 11 60 8 480

Give the multiplicative inverse of −2 7 = . 7 −2

Multiplicative inverse of Do It Together

72 − 35 60

8 11

−2 . 7

Fill in the blanks using the properties of rational numbers. 1

1 5 3 − ÷ = 4 8 2

6 × 8

= 0

−5 and 5 8

−12 and 4.2 7

23 and −3.2 5

−13 by 3.6 6

34 51 by 7 19

−

3 ÷ 2

38 16 and 7 −19

5 14 by 5 3

−28

× 1 =

13 18

Do It Yourself 1C 1 Multiply. a e

and

18 7

2 Divide.

30 −7

13 and −2.5 15

−51 and −28 5

12 25 by 5 −4

31 by −7 14

−36

and

−27 44

by 4.8

65 −14

3 If the product of two rational numbers is 28 and one of them is −17 , find the other. 3

4 Fill in the blanks. a d

−9 −7

5 If a =

5 4

8 ×

× −6 5

,b=

−7 9

÷ 3 8

b −4.5 ÷

____

−13

____

and c =

25 4

16 5

____

÷ 1.5 =

____

−8

f 3.4 ÷

−7 3

14 5

____

−4 = ____ 9

then verify

a a × (b × c) = (a × b) × c

b (a − b) ÷ c = a ÷ c − b ÷ c

c a×c=c×a

d (a − b) − c ≠ a − (b − c)

6 Sunita has a square plot of length 185 m. What is the area and perimeter of the plot? 3

7 A ribbon of length 357 m has been cut into 24 equal pieces. What is the length of each piece? 8

Chapter 1 • Rational Numbers

8 A zoo ticket usually costs ₹ 250 for elders. For kids, they are priced at 4

for 5 kids’ tickets?

5 8

of the usual cost. How much will it cost

9 The area of a room is 389 m2. If its length is 247 m, find its breadth. 4

10 Simplify:

5 2 9 31 17 34 11 ÷ × ÷ × + 8 8 7 13 15 21 7

11 During the summer holidays, Mansi read a book. After reading pages are there in the book?

5 8

of the pages, 120 pages were left. How many

Word Problems 1 2

Kavya bought buys 8 rolls of paper towels. The total length of all the rolls was is 88 is the length of 5 such rolls?

1 8

m. What

2 3 Kapil earns ₹18,000 per month. He spends of the total income on rent, of the remaining on 9 7 1 food and of the rest on shopping. How much will he be able to save in a year? 4

Points to Remember • A rational number is a number that is expressed as p , where p and q are integers and the q denominator should never be equal to zero. A negative rational number is always less than a positive rational number. • To compare two or more rational numbers, make their denominators the same and compare the numerators. •

There is an infinite number of rational numbers between two rational numbers.

• The operations on rational numbers are similar to fractions. The signs need to be kept in mind like we do in integers.

Math Lab Setting: In groups of 4 Materials Required: Cards with properties of operations written on them, paper and pen. Method: • Prepare cards with different properties of operations written on them for example, 1 3 5 1 5 3 5 + ÷ = ÷ + ÷ 2 5 7 2 7 5 7 •

Distribute an equal number of cards among the groups.

• Each group identifies the property and writes its name along with the solution in on the paper. •

The group that finishes all the sums first correctly first, wins!

Chapter Checkup 1 Which rational number does point P represent on the given number line? P

−2

−1

2 Give a rational number equivalent to −9 having 87 as the denominator. 29

3 Compare and fill in the boxes. a

−5 7

12 32 e −16 7.2 f 5 8 −6 Arrange the given rational numbers in descending order. a 1.2,

5 2

, −1

1 −7 15 , , 2 6 6

−12

, 1.75, −0.85, −1

−12 7

−8 3

6 Give four rational numbers between 7 If m =

c −12 5

and

1 −3 8 ,n= and o = then verify 6 5 10

a m ÷ (n ÷ o) ≠ (m ÷ n) ÷ o

18 4

21 19

1.75

29 3

51 6

5 Circle the rational numbers that do not lie between −2 and −1. a

−15

18 5

, −5, 3.25, 2

−10

3 21 , 6 5 −15 8

b m × (n + o) = m × n + m × o

c m−n≠n−m

d (m − n) + o = m + (n + o)

8 Fill in the blanks. a The sum of a rational number and its additive inverse is __________. b Reciprocal of a negative rational number is __________. c The rational number __________ does not have any reciprocal. d Adding __________ to a rational number gives the rational number itself.

9 The sum of two rational numbers is −

8 5

10 Which rational number should be subtracted from

12 7

to get

−15 4

11 Find the area of a triangle with a base of 156 m and a height of 12 Solve and find the multiplicative inverse. 7 −2 + 5 6

e −13 ÷ 25 26 5

Chapter 1 • Rational Numbers

3 7

19 17

5 14 −51

9 2

−

17 6

191

8 +

9 , find the other. 17

, if one of the numbers is

−8 3

24 59

−12 7

−21 4

h −24 × −29 13 39

13 One roll of ribbon is 15

2 3

m long. What is the length of 5 full and one-fourth of such ribbons?

14 Find (m + n) ÷ (m − n), where m =

5 6

and n =

−8 9

15 The given graph shows the time taken by different runners in a 500 m race. What is the average time taken by all the runners to complete the race in minutes?

Time taken in sec

105 100

100.26

91.63

90.75

90 85

98.55

Satish

Itisha

Rohan

Rhea

Runner

16 Simplify

12 7

−5 9

2 8

7 5

−

Between which two whole numbers does the result lie?

−5

Word Problems 1

2 3

Rohit practiced practised cricket for 12

5 1 hours last week. He practised 2 hours this week. 6 8

For how many hours did Rohit practice practise cricket this week? 1 Nikita distributed 52 kg of rice equally among 15 families. How much rice did each family 4 get? 256 175 Vivek has a rectangular plot of length m and width m while Shweta has a square 5 3 189 plot of length m. Whose plot has a larger area? Also, find the difference in between the 2 areas of both the plots.

Richa bought 1 bought 1

3 5 packets of chips, Alex bought more packets than Richa and Prince 4 6

1 times as many packets as Richa. What is the total number of packets of chips 2

bought by all of them?

3 3 , Ronita divided it by . If the difference between the 5 5

Instead of multiplying a number by

wrong and the correct answer is 80, then find the number. 2 4 Shilpa had ₹750 with her. She spent of the money on purchasing a book and of the 5 9 remaining amount on stationary. How much money is left with her?

Solving Equations in One Variable

2 Let's Recall

Neeta is learning about the ages of all the family members in her family. She asks her mother about her brother’s age. She says that she is 5 years more than 5 times her brother’s age. She also tells her that she is 40 years old. Neeta wants to calculate her brother’s age. We know how to solve the equations of the type ax + b = c by the balancing method and transposing method. Let us recall the two methods. Balancing Method

Transposing Method

5x + 5 = 40

Subtracting 5 from both sides of the equation, we get,

Transposing 5 from the LHS to the RHS, we get, 5= x 40 − 5

5 x + 5 − 5 = 40 − 5

Simplifying the RHS of the equation, we get,

Simplifying both sides of the equation, we get,

5 x = 35

Transposing 5 from the LHS to the RHS, we get,

Dividing 5 on both sides of the equation, we get,

x =7

5 x 35 = 5 5 Simplifying both sides of the equation, we get, x =7

So, Neeta’s brother is 7 years old.

Letʼs Warm-up

Match the linear equations with their solutions. Linear Equation

Solution of the equation

1 2x + 4 = 8

x=7

x = −6

3x = 9 2

3 3x − 6 = 15

x=6

4 −5x = 40

x=2

−

x = 3 2

x = −8

I scored _________ out of 5.

Mean, Median Mode Solving Linearand Equations Real Life Connect

Vimala and Jaya are raising money by selling tickets for a hockey tournament. Both Vimala and Jaya have sold the same number of tickets. Jaya has sold thirty-six tickets more than Lata. Vimala has sold three times as many as Lata has. So, how many tickets has Lata sold?

Transposing Method Let us denote the number of tickets sold by Lata be x. So, the number of tickets sold by Jaya are 36 + x. And the number of tickets sold by Vimala are 3x. Now, according to the situation, Vimala and Jaya sold the same number of tickets. 3x = 36 + x

Let us now solve this equation using different methods. Balancing Method Imaging the expressions on two sides of a simple balance.

For the balancing method, we perform the same mathematical operations on both sides of the equation, so that the balance is not disturbed. The equation is 3x = x + 36 Subtract x from both sides, 3x − x = x + 36 − x ⇒ 2x = 36 Divide both sides by 2,

2x 36 = ; ⇒ x = 18 2 2

Thus, Lata sold 18 tickets. Solve: 5x + 21 = 2x + 39 5x + 21 – 21 = 2x + 39 – 21 5x = 2x + 18 ⇒ 5x – 2x = 2x + 18 – 2x 3x = 18 ⇒ x=6

3 x 18 = 3 3

Let us understand some important terms, Equation: 2x + 2 = 6 + x An expression is a combination of terms (variable or numbers or both) that are connected using mathematical operations.

An equation is a mathematical statement with an equal symbol between two expressions. 2x + 2 Expression

6+x Expression

Linear equation in one variable is an equation that has linear expressions with only one variable. Variable

LHS: 2x + 2

Equal to symbol

Equation

RHS: 6x + 2

Solution of a Linear Equation The method of finding the value of the variable so that it satisfies the equation is called solving a linear equation. The value of the variable obtained from the method is called the solution of the linear equation. Let us understand the transposition method. Transposition is the process of shifting a term of an equation to the other side simply by changing its ‘sign’ or by applying an opposite operation. When any term of an equation can be taken to the other side of the equation with the change of its sign then this method is called transposition. Solve for x: 5x + 12 = 3x + 4 Steps for the transposition method:

Step 1

Step 2

Step 3

Identify the variable and constants

Remove the brackets to simplify the LHS and RHS.

Move all variable terms on one side and all the constant terms on the other side.

5x + 12 = 3x + 4

No brackets here

Variable: x Constants: 12, 4

5x – 3x = 4 – 12

Step 4 Simplify the LHS and RHS to solve for the variable. 2x = – 8 x = –4

Remember! While transposing, keep the variable terms on one side of the equation and the constants on the other side.

Chapter 2 • Solving Equations in One Variable

Example 1

Solve using the balancing method. 1

3x + 47 = 95

5x + 15 = 3x + 33

3x + 47 – 47 = 95 – 47

5x + 15 −15 = 3x + 33 − 15

3x = 48

5x = 3x + 18 ⇒ 5x − 3x = 3x + 18 − 3x

x = 16

2x = 18 ⇒ x=9

2 x 18 = 2 2

Remember! The mathematical operation is reversed when we move terms from one side to the other side of the equal symbol.

Example 2

Solve using the transposition method. 1

Do It Together

8x – 14 = 3x + 11

Error Alert!

6x – 25 = 4x – 13

8x – 3x = 14 + 11

6x – 4x = 25 – 13

5x = 25

2x = 12

x = 5

x=6

Add: x + 3 = 7 On solving the equation:

x=7+3 x = 10

Find the value of x. 2

3x + 4(x – 1 ) = 10  3x + __________________ = 10  __________________  __________________  __________________

Do It Yourself 2A 1 Find the value of the unknown weights. a

2 Find the value of x using the balancing method.

a 6x + 4 = 22

b 2x + 3 = x + 18

c 7x + 45 = 4x + 78

d 2x + 17 = 9x – 32

e 11x + 1 = 3

f 5x – 11 = 2x + 7

x=7–3 x=4

3 Find the value of x using the transposition method. a 3x – 5 = 2x + 97

b 5(x – 7) – 2x = 13 – x

c 15x – 2(18 + x) = 29

d x + 2(3 + x) = 5(x – 6)

e 9x + 3 = 13(2x + 8)

f 5(x –17) = 7(x –19)

4 Solve for the value of x in the following equations. a 3x + 4(x – 2) = 41

b x – 1.5(x – 3) = 2(x − 3)

c 3(1.5x + 1) = 16.5

d 2.4x + 4.8 = 1.2x + 6

e 4x + 7 = 1.5(x + 12)

f 1.3(x – 3) + 2.6 = x

5 One number is twice another number. If 30 is subtracted from both numbers, then one of the new numbers becomes four times that of the other new number. Find the numbers.

6 Find two consecutive even integers so that two-fifths of the smaller number exceeds two-elevenths of the larger number by 4.

7 One number is 3 times the other number. If 15 were to be added to both the numbers, then one of the new numbers becomes twice that of the other new number. Find the numbers.

8 Solve for the value of x. 18 cm

x Perimeter = 54 cm 12 cm 15 cm

b x+3

2x + 3

c 4

Perimeter = 56 x – 1

Perimeter = 35 x+4 x+3

2(x + 1)

3x – 5

Word Problems 1

After 30 years, Monty’s age will be 6 more than thrice his present age. What is Monty’s

Harsh’s mother is 30 years older than he. In 12 years, Harsh’s mother will be three times as

The length of a rectangular field is 7 less than twice its breadth. If the perimeter of the

Ravi deposited ₹25,000 in his bank account. He deposited the currency notes of ₹50, ₹100

The ratio of Naveen’s age to his father’s age is 2:5. After 10 years, the ratio of their ages is

present age?

old as Harsh. What are the present ages of Harsh and his mother? rectangle is 196 m. Find the area of the field.

and ₹500 in the ratio of 6:2:1. Find the number of notes of each denomination. 1:2. What was Naveen’s father’s age at the time of Naveen’s birth?

Simplifying Equations to Linear Form Remember, Lata, Jaya and Vimala sold hockey tickets. They sold the tickets online as well as offline. The cost of the online ticket was ₹200 more than that of the offline ticket. Five-eighths of the online ticket’s cost is the same as three-fourths of the offline ticket’s cost. Let us find the cost of each type of ticket. Let the cost of the offline ticket be ‘y’.

Chapter 2 • Solving Equations in One Variable

5 3 ( y + 200) = y 8 4

1 y = 125 8 y = 1000 y + 200 = 1200

5 3 y + 125 = y 8 4 3 5 y− y= 125 4 8

Thus, the cost of the offline ticket is ₹1000 and the cost of online ticket is ₹1200. Solve for x. x +8 3

x +3 2

Step 1: M ultiply the numerator of the LHS Step 2: S implify both sides of with denominator of the RHS. x+8 3

the equation.

Step 3: A pply transposition method to find the value of x.

2x + 16 = 3x + 9

x+3 2

16 – 9 = 3 x – 2 x x =7

2( x + 8) = 3( x + 3) Example 3

2 x + 7 5x + 4 = 3 5 5(2x + 7) = 3(5x + 4) Solve for x:

Example 4

Solve:

10x + 35 = 15x + 12

22 x − 143 − 5 x + 15 = x −9+ 5 11

35 – 12 = 15x – 10x 23 = 5x x=

Do It Together

(2x − 13) x − 3 x − 9 = +1 − 5 11 5 ( x − 9) + 5 11(2 x − 13) − 5( x − 3) = 5 × 11 5

17 x − 128 = x −4 11 17x − 128 = 11x − 44 ⇒ 17x − 11x = 128 − 44

23 5

Solve for m:

3x + 5 1 = 2x + 1 3

6x = 84 ⇒ x = 14 3(3x + 5) = 1(2x + 1)

Do It Yourself 2B 1 Write YES if the correct solutions are given for the linear equations, else write NO. a x = 2 for

5x = 3 2x − 1

2 Solve. a d

5x − 4 7

8 9

6 9 = 3n + 1 5n + 3

b x = 6 for

b e

z + 5 6

−

(4x + 1) 3

7 5 = x +1 ( x − 1)

z +1 9

c x = 3 for

z + 3 = 4

(2 x − 1) 2

−

(3 x − 7) 5

6 =

7x + 1 3

y − (4 − 3 y )

2 y − (3 − y )

1 4

5x + 6 4

3 Find the value of x using the cross-multiplication method. a d

3x − 8

= 6

5(2 − x ) − 4(1 + x ) 2 − x

5 8

4 Answer the given questions. a Solve:

3x − 1 2

4x + 6 3

5 7 = x x + 2

6 9 = 3x + 1 5x + 3

0.4x + 5 8

2.5 x + 4 15

2 3 43 + = 2 and find the value of y if = x y 6

2 x 4x − 5 3 x + 5 + = 11 and find the value of y if = 3 y 11 10 What is the postal code? b Solve:

The numerator of a fraction is 3 less than the denominator. If the numerator and the denominator both are decreased by 2 then the simplest form of the fraction becomes

3 . What is the fraction? 4

Two complementary angles are in the ratio 1:2. When 15 is added to the smaller angle and 15 is subtracted from

Solve the values of the variables to decode the postal code: 2abc5d.

the bigger angle, the new ratio of the angles became 1:1. Find the angles.

a −1

7a − 27

1 3

2b + 6 4

7b − 34 3

11c − 44 5

3c + 17 4

9d − 7

7d + 3

29 31

Word Problem 1

The present ages of Abhijeet and Sahil are in the ratio 2:1. Four years ago, the ratio of their ages was 5:2. Find the present ages of Abhijeet and Sahil.

Application of Linear Equations Remember Lata and her friends sold hockey tickets. Jaya sold 54 tickets in total. If she earned a total amount of ₹58,800 and she sold two types of tickets, one for ₹1000 and the other for ₹1200, then how many tickets of each type did she sell? Linear equations help us in solving real-world problems. We need to set up an equation according to the condition given and solve it to find the value of the unknown. This method consists of two steps: 1

Translating the word problem into symbolic language.

Solving the equation

Let us solve a few real-life problems. Let the tickets sold for ₹1000 be x. Then the tickets sold for ₹1200 will be 54 – x.

Chapter 2 • Solving Equations in One Variable

According to the question, 1000x + 1200(54 − x) = 58800 ⇒ 1000x + 64800 − 1200x = 58800 200x = 64800 − 58800 ⇒ 200x = 6000 x = 30 ⇒ 54 − x = 54 − 30 = 24 Thus, Jaya sells 30 tickets for ₹1000 and 24 tickets for ₹1200. Example 5

Suman’s present age is equal to one-fifth of her mother's age. Twenty-five years later, Suman’s age will be 4 years less than half the age of her mother. Find their present ages. y Let us assume the mother’s present age to be y years; Suman’s present age = 5 According to the condition, y + 25 = 5

y + 25 −4 2

Multiply both sides by 10, 2y + 250 = 5(y + 25) − 40 ⇒ 2y + 250 = 5y + 125 − 40 5y − 2y = 250 − 85 ⇒ 3y = 165 ⇒ y = 55 Suman’s present age =

y 55 = = 11 years 5 11

Thus, Suman’s present age is 11 years and her mother is 55 years old.

Example 6

A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers? Let us assume the first number to be x; second number = 5x According to question, 5x + 21 = 2(x + 21)

5x + 21 = 2x + 42 ⇒ 5x − 2x = 42 − 21 3x = 21 ⇒ x = 7 ⇒ 5x = 35

Thus, the numbers are 7 and 35. Example 7

If the sum of three consecutive multiples of 9 is 108, then find the second multiple. Let us assume the smallest multiple of 9 to be 9x. Next multiple = 9(x + 1); Last multiple = 9(x + 2) So, 9x + 9(x + 1) + 9(x + 2) = 108 27x + 27 = 108 ⇒ 27x = 81 ⇒ x = 81 ÷ 27 ⇒ x = 3 So, we can find, 9x = 27, 9(x + 1) = 36, 9(x + 2) = 45 Thus, the three consecutive multiples of 9 which add up to 108 are 27, 36, 45.

Example 8

Tanya attempts a test in which there are 60 questions. For each correct answer x marks will be awarded and for each incorrect answer one-fourth of correct answer marks will be deducted and 0 marks will be awarded for un-attempted question. If Riya attempts 45 questions out of which 29 are correct, what is the value of x if Riya is awarded 200 marks?

Marks awarded for each correct answer = x Marks deducted for each incorrect answer = Total number of questions attempted = 45

x 4

Did You Know?

Total number of correct answers = 29

Knowledge of algebraic expression

Total number of incorrect answers = 45 – 29 = 16

days, as we are often estimating

Total marks scored = 200

can help us plan and schedule our and solving for unknown variables.

 x 200 So, (29 × x ) −  16 ×  = 4  29x − 4x = 200 ⇒ 25x = 200 x=

200 ⇒x=8 25

Thus, the value of x is 8 marks. Do It Together

A car rental company charges ₹500 per day plus ₹12 per km. If the bill is ₹3320 for a day, then how many km were driven? Let the distance travelled be x km. Fixed charges for a day = __________________ Rate of the car for each km = __________________ Total bill amount = __________________

Do It Yourself 2C 1

Form the equation for the cases given. a Thrice x added to 5 is equal to the difference of half of x and 13. b The sum of four times y and 11 is equal to the sum of two times y and 56. c The sum of three consecutive even numbers is 144. d A train travels from point A to point B, a distance of 240 miles. On the return trip, the train’s speed is increased by 20 mph and takes 2 hours less.

The sum of four consecutive odd numbers is 352. What are the numbers?

The sum of three consecutive multiples of 11 is 429. What are the numbers?

A number is decreased by 10% and the new number obtained is increased by 25%, so the result is 81. Find the

The angles of a triangle are in the ratio 3:4:5. What is the difference between the largest angle and the smallest

Divide 6000 into two parts so that 15% of one number is equal to 35% of the other number.

number.

angle?

Chapter 2 • Solving Equations in One Variable

Puneet’s age is three times the age of his daughter. If the sum of their ages is 52 years, find the difference

The sum of two numbers is 45 and the numbers are in the ratio 2:1. What are the numbers?

The total cost of a table and a chair is ₹15,550. The cost of the table is ₹550 more than the cost of the chair. What

between their ages.

is the cost of the table?

10 Virat scores 20 more runs than twice the runs scored by Rohit. Together, their runs are four runs short of a triple century. What are the individual scores of Virat and Rohit, respectively?

11 Amit has a certain number of pencils that cost ₹4 per pencil. He again purchased 10 more than twice the number of pencils he had. If the total cost of all the pencils is ₹290, how many pencils did Amit have initially?

12 Animesh left one-half of his estate to his daughter, one-third of his estate to his son and donated the rest of his estate. If the donation was ₹5,00,000 then how much estate did, he give to his daughter and son?

13 One of the two digits of a two-digit number is 40% of the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 77. What is the original number?

14 The speed of the flow of a river is 3 km per hour. A boat goes upstream between two points in 5 hours while it covers the same distance going downstream in 3 hours. What is the speed of the boat in still water?

15 A sum of ₹770 is made up of denominations of ₹5 and ₹10. If the total number of notes is 72 then find the number of notes of each denomination.

16 Two cars simultaneously start from A and B in opposite directions and the distance between them after 4 hours is 60 km. What is the speed of each car?

's' km per h

('s' + 10) km per h A

60 km

300 km

Points to Remember • An equation is a mathematical statement with an equal symbol between two expressions. • Any operation done on one side of an equation must be done on the other side of the equation so that the equation remains true. • We use the transposition method to collect similar terms on one side. Change the signs of terms while using the transposition method and moving terms from one side to the other side. • While solving the word problems, identify the unknown terms, form an equation and solve the value of the unknown term. • The following rules can be applied while solving linear equations.

The same number can be added or subtracted from both sides of the equation.

The same number can be multiplied or divided on both sides of the equation.

A term can be transposed to the other side of the equation with its sign changed.

Math Lab Aim: Solve linear equations in one variable Materials required: Chalk, stopwatch and equation cards Setting: In groups of 4 Method:

Prepare equation cards (unsolved equation on cards (should have linear equations)).

Mark the starting line and finishing line in the classroom or in an open area. There could be 2 different laps.

Each team lines up behind the starting line and the first person of each team is given an equation 3 card.

The student solves the equation and runs towards the finishing line with the correct solution.

Once they reach the finishing line, they hand the card to the next teammate. The next person 5 solves the next equation and continues the relay.

Use a stopwatch to calculate the total time taken by each team to complete the race.

The team which has the fastest completion time wins the relay race.

Chapter Checkup 1

Find the value of unknown value of x. a

Solve using the balancing method. a 4x + 14 = 56

b 3x + 4 = x + 18

c 5x – 7 = 2x +11

d 3x – 4 =1 – 2x

e 5x + 48 = 3(4x – 5)

11(x + 1) = 12(x – 1)

Solve using the transposition method. a 5(x + 3) = 3(1.5x + 18)

b 6(2x + 11) = 8(2x – 1)

c 8x – 7 – 3x = 6x – 2x – 3

d 10x – 5 – 7x = 5x + 15

e 5(x – 1) = 2(x + 8)

4x – 3 = (3x + 1) + (5x – 4)

Which of these is the correct solution of the equation? a 5x + 7 = 35 for x = 7

b 3(x – 2) = 4(x + 3) for x = 18

d 4(x + 1) – 3(2x – 18) = 2x – 2 for x = 15

Chapter 2 • Solving Equations in One Variable

c 4x + 3 – 2(x + 1 ) = 8x for x =

e 5(x + 9) – 2x = 90 for x = 15

1 6

Find the value of x using the cross-multiplication method. a

3x 9 = 2( x + 2) 10

4( x + 2) + 3 27 = 2 x + 3( x − 1) 17

4( x + 3) − 4 3x + 1 = 5 8

11(8 − x ) − 3 7( x − 3) + 2 = 7

3m + 2 3 = 9m + 4 5

5 z + 14 =3 2z

7 3 5w + = w − 14 2 2

3 b+7 5 4 = 2 4 b−4 5

11(2 x + 1) 11

4( x + 6) 26

= 6 x − 4 − 3x

Solve the equations. a

2 y − 1 17 = y + 3 12

2 5 1 − = 5a 3a 15

Answer the questions. F a If=

5 x + 11

9 C + 32, then find C when F = –40. 5

b Solve:

x+y 3 9 x + 4 4(2 x − 1) = . = and then find the value of y if xy 8 10 7

c Solve:

2 5 2x + 1 x − 2 3 + 2x = 4. − = and then find the value of m if + x m 10 6 15

Find the value of the unknown variables to decode the phone number. 8ab00c1d2e

a 9(x + 1) − 4(x + 2) = 6

b 11(x − 2) = 2(x + 1) + 3

2x + 7 1 = 3(7 x − 8) 4

11 x − 3(2 x − 5) 3

7 x + 15

2( x − 2) 1 = 3(3 x − 5) 3

Place the correct mathematical operations (+, –, ×, ÷) so that the equations are true. a 2x __ 3 = 7 for x = 2 for x = 2 c

2x __ 3(x __ 1)

15x __ 2(2x __ 4)

18 for x = 3 25

b 3(x __ 1) __ 4x = 46 for x = 7 d

3x __ 4(x __ 3) x __ 2(3x __ 7) = for x = 4 20 7

10 Two numbers are in the ratio 4:9 and the difference between the numbers is 250. What is the sum of the numbers? 1 1 1 from a number and multiply the result by then you get . What is the number? 11 If you subtract 2

12 Three consecutive even numbers add up to 696. What are the numbers? 13 The numerator of a fraction is less than its denominator by 8. If both the numerator and the denominator are increased by 17 then the fraction obtained is

5 in its simplest form. Find the original fraction. 6

Word Problems 1 The ages of Kushagra and Kush are in the ratio 5:6. Five years later the sum of their ages will be 43 years. What is the difference between the ages of Kushagra and Kush?

2 The present age of Shagun is one-fifth of her mother’s age. After twenty-five years, her age will be 4 less than half of her mother’s age. Find the difference between their present ages.

3 The perimeter of a rectangular swimming pool is 170 m. If the length of the pool is 4 m more than the breadth of the pool, what is the difference between the length and the breadth?

4 Sheetal has currency notes of denominations of ₹50, ₹100 and ₹500. The ratio of the number of notes is 12:8:7 respectively. If the total amount of money with Shetal is ₹24,500 how many notes of each denomination does she have?

5 There is a rectangular plot for a school. The length and breadth of the plot are in ratio

15:7. At the rate of ₹150 per metre, it will cost ₹1,32,000 to fence the plot. What are the dimensions of the plot?

6 Vikram took goats to the field. Half of the goats were grazing. Three-fourths of the

remaining goats were playing nearby. The rest of them, 15, were drinking water from the pond. How many goats were there in total?

7 The ratio of the speed of a boat to the stream is 7:2. The boat takes 6 hours more travelling upstream than downstream. What is the time taken by the boat for the entire journey?

8 The angles of a quadrilateral are in the ratio 7:17:19:29. What is the measure of the angles? 9 A book has 400 pages. Raj finishes writing 100 pages in x minutes. Raju takes twice the time taken by Raj to write the next pages. Vivek takes 2 hours more than half the time taken by

Raju to write the remaining pages. If the total time taken to write the book was 33 hours and 20 minutes then what is the value of x?

10 ₹x is to be divided among three friends A, B, and C. The share of A is two-fifths of the total money, the share of B is two-thirds of the remaining money and C’s share is ₹600. What is the value of x?

11 A father is 20 years older than his son. 12 years ago, the age of the father was six times that of his son. Find their present ages.

12 The age of the father is three times the age of his son. After five years, the age of the father will be 2

1 times the age of his son. Find the present ages of father and son. 2

Chapter 2 • Solving Equations in One Variable

3 3 Polygons Let's Recall

Shapes can be made with lines, curves or both lines and curves. A curve is a line that is not straight. It can bend and twisted in any direction. Curves can be open, closed, simple or non-simple. Open Curve

Closed Curve

Simple Curve

Non-simple Curve

Shapes made with only line segments are called polygons. Let us see some polygons and non-polygons: Polygons: Closed

Non-polygons: Open

The region inside a closed shape is called the interior. The region outside the closed shape is called the exterior. Exterior

Letʼs Warm-up

Interior

Name the shapes as open curves, closed curves, polygons, non-simple curves. 1

__________________ 3

__________________

__________________ 4

__________________ I scored _________ out of 4.

Understanding Polygons and their Properties Real Life Connect

Mr. Singh owns a tile shop. He sells tiles in a lot of different shapes like triangles, rectangles, hexagons etc.

We know that these are called 2-D shapes. Some two-dimensional shapes are made up of straight lines and connected at their endpoints. These are called polygons. Now, imagine if all of these shapes were made of curves. Could we tile them? A polygon is a simple closed curve made up of 3 or more sides, with no overlapping or self-intersecting parts. Polygons

Remember! In a polygon, the number of sides is always equal to the number of vertices.

Non Polygons

Did You Know? The term “polygon” comes from the Greek words “poly,” meaning many, and “gonia,” meaning angles.

Think and Tell In the given polygon ABCDE, AB, BC, CD, DE and EA are the sides. A, B, C, D and E are the vertices. ∠A, ∠B, ∠C, ∠D and ∠E are the angles. Let us see learn some more terms.

Chapter 3 • Polygons

Which polygon do you see most often in your everyday life?

Diagonals are line segments joining opposite vertices of a polygon. E.g. - AC, AD C

Interior of a Polygon is the region enclosed by the sides of the polygon.

Adjacent Sides are sides that share a common vertex (corner). They are next to each other in the sequence of sides. E.g. (AB-BC, BC-CD, CD-DE, DE-EA)

Exterior of a Polygon is the region outside the sides of the polygon.

Adjacent Angles are angles that share a common side (arm). They are next to each other in the sequence of angles. E.g. (∠A-∠B, ∠C-∠D, ∠D-∠E, ∠E-∠A)

Convex Polygon: In these polygons measure of each angle is less than 180º. Concave Polygons: In these polygons measure of one of the angles is more than 180º. Polygons can be convex or concave Regular Polygon: In these polygons the length of all the sides are equal.

Polygons can be regular or irregular

Irregular Polygon: In these polygons the length of all the sides are not equal.

Classifying Polygons Polygons can be classified into different types on the basis of their number of sides and vertices.

Remember! A polygon has to have at least 3 sides.

Triangle - 3 sides

Quadrilateral - 4 sides

Pentagon - 5 sides

Hexagon - 6 sides

Heptagon - 7 Sides

Octagon - 8 Sides

Nonagon - 9 Sides

Decagon - 10 Sides

We know that a diagonal of a polygon is a line segment that joins two vertices of the polygon, which are not already joined by the adjacent vertices of the polygon. Let us draw the diagonals in some polygons.

Polygon Triangle

Pentagon

Octagon

Decagon

Number of sides (n)

Number of Diagonals

3 (3 − 3) =0 2

5 (5 − 3) =5 2

8 (8 − 3) = 20 2

10 (10 − 3) = 35 2

Diagonal

If a polygon has n number of sides, then, the number of diagonals n (n − 3) . 2 In a convex polygon, all the diagonals lie inside the polygon, whereas, in a concave polygon, at least one diagonal lies outside the polygon. Diagonals in a

concave polygon Diagonals in a

convex polygon Example 1

Find the number of diagonals in a nonagon. Number of sides in a nonagon (n) = 9

Think and Tell

Number of diagonals in a nonagon =

polygon with three sides?

n (n − 3) 9 (9 − 3) = 27 = 2 2 Therefore, a nonagon has 27 diagonals. Example 2

Identify the given polygons as concave or convex. •

•

Concave Polygons

Convex Polygon Do It Together

Is it possible to have a concave

Draw the diagonals in the given polygons. Find the number of diagonals using the formula. Identify the polygon as convex or concave.

Polygon

Number of sides (n) Diagonals

5 9

Convex/Concave Chapter 3 • Polygons

Do It Yourself 3A 1 Fill in the blanks. a A polygon with 7 sides is called a __________________. b The number of diagonals in a hexagon is __________________. c A polygon with all sides and angles equal is called __________________. d A polygon with 10 sides is called a __________________. e A regular polygon is a 2D shape where the sides are all __________________ line segments of __________________.

2 Write True or False. a A triangle is a convex polygon. b The number of diagonals in an octagon are 7. c A circle is an example of a 2-D geometrical shape that is not a polygon. d A rectangle is a regular polygon.

3 Classify the following as convex or concave polygons based on the angles of the polygons. a 65º, 115º, 111º, 70º, 179º

b 176º, 105º, 110º, 66º, 195º, 165º, 83º

4 Classify the polygons as concave or convex. a

5 A polygon has 6 sides and all the angles in the polygon measure 120°. Is this a convex or concave polygon? 6 Draw a hexagon and draw all the diagonals in the polygon. 7 Use the formula to find the number of diagonals in polygons with the number of sides given. Name the polygon. a 4 sides

b 8 sides

c 10 sides

8 In a regular octagon, how many diagonals can be drawn from one vertex? 9 How many non-overlapping triangles can we make in a polygon having 5 sides by joining the vertices?

Word Problem 1

Rajan drew a regular polygon with 7 sides. Name the polygon. He then shifted one of the vertices inside the polygon. How many diagonals does this polygon have? Identify the polygon as concave or convex.

Angle Sum Property of Polygons

Exterior Angle

We know that an angle is formed when two straight lines meet at a common vertex.

I nt An erior gle

• Any angle that is formed in the polygon is called an interior angle. • The angle between a side of a polygon and an extended adjacent side and is formed outside the polygon is called an exterior angle.

The sum of all the interior angles of a triangle is 180°. What about the polygons with four or more sides? Since a triangle has three sides, we find the measurements of the angles accordingly. Let us find the sum of angles of any polygon. Step 1: Mark one of the vertices in the polygon. Step 2: Draw diagonals from that vertex to form triangles. Count the number of triangles. Step 3: Since the sum of angles in a triangle is 180°, multiply 180° by the number of triangles formed inside each polygon. 4

1 Pentagon

Quadrilateral

Hexagon

Heptagon

Number of sides

4 sides

5 sides

6 sides

7 sides

Number of Triangles

180° × 2 = 360°

180° × 3 = 540°

180° × 4 = 720°

180° × 5 = 900°

From this we can deduce that the sum of interior angles of a polygon is the product of 2 less than the number of sides of a polygon. This can be written as: Sum of interior angles of a polygon = (n - 2) × 180° where n is the number of sides of the polygon. The angle sum property doesn’t depend on whether the polygon is a concave polygon or a convex polygon. Let us find the measure of the missing exterior angle in the polygon. The shape has 6 sides and is therefore a hexagon.

126°

Sum of angles in a hexagon = (6 - 2) × 180 = 720°

126°

Let the missing angle be x

106°

720° = 126° + 126° + 106° + 129° + 147° + x

720° = 634° + x

147°

So, x = 720° - 634° = 86° Example 3

Find the sum of interior angles in a convex octagon using the angle sum property. Number of sides in a convex octagon n = 8 Using the angle sum property, Sum of interior angles = (8 - 2) × 180° = 1080°

Example 4

Find the interior angle sum in a nonagon. Number of sides in a convex octagon n = 9 Using the angle sum property, um of interior angles = (2(9) - 4) × 90° = S (18 - 4) × 90° = 1260°

(9 - 2) × 180° × Chapter 3 • Polygons

129°

2 2

= (2 × 9 - 2 × 2) × 180° = (18 - 4) × 90° 2

Do It Together

Find the sum of angles in the figure. Find the measure of the missing angle. The shape has 7 sides so it is a _____________. Sum of angles of the _____________ = (n - 2) × 180 Measure of the missing angle is _____________.

138° 143°

67°

152° 108°

? 134°

Do It Yourself 3B 1 Fill in the blanks. a A quadrilateral has _______ as the sum of its interior angles. b The sum of interior angles of a polygon of n sides is _______ right angles. c Each interior angle of a regular hexagon is _______. d If the number of sides of a regular polygon is _______, then the interior angle sum is 1800°.

2 Write if true or false. a The sum of the interior angles of a triangle is 270°. b The number of triangles that can be formed within a polygon with n sides is n. c The maximum measure of the interior angle for a regular polygon is 180°. d It is possible to have a polygon whose sum of interior angles is 540°. e If each interior angle of a regular polygon measures 168°, the polygon has 30 sides.

3 Find the interior angle sum of a polygon with: a 7 sides

b 13 sides

c 17 sides

d 20 sides

e 22 sides

4 Is it possible to construct a polygon, the sum of whose interior angles is 20 right angles? If yes, find the number of sides of the polygon.

5 One of the angles of a polygon is 100° and each of the other angles is 110°. Find the number of sides in the polygon.

Word Problem 1

To manufacture a tile, the worker needs to determine the measure of each interior angle of the tile. What is the measure of each interior angle if the tile is a regular octagon?

Exterior Angle Sum Property of Polygons The angle between a side of a polygon and an extended adjacent side and is formed outside the polygon is called an exterior angle. The sum of exterior angles in any polygon is always a constant, i.e. 360°. Sum of Interior angle + Exterior angle = 180°.

The angle formed on a

∴ Exterior angle = 180° - Interior angle

Let us calculate the measure of exterior angles to see what the total is. We see that the sum of exterior angles in any polygon is 360°. 53°

38°

106°

71°

88°

Remember!

60°

59°

50°

straight line is always 180°.

47°

111° 37°

Sum of exterior angles of a hexagon

60° + 71° + 53° + 88° + 38° + 50° = 360°

111° + 47° + 106° + 59° + 37° = 360°

Similarly, if a polygon has n number of sides, then, the measure of each exterior angle of a regular 360º . n Find the measure of each exterior angle in the figure. polygon = Example 5

We know that the exterior angle sum = 360°

(3x − 5)°

→ ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = 360°

→ (2x - 1)° + (3x - 5)° + (8x + 3)° + (7x - 2)° + (4x + 1)° + (6x + 4)° = 360° → 30x = 360° → x = 12°

F (6x + 4)°

C E (7x − 2)° D (4x + 1)°

∠C = (8x + 3)° = 99°; ∠D = (7x - 2)° = 82°; ∠E = (4x + 1)° = 49°; ∠F = (6x + 4)° = 76°

The exterior angle sum of a 30-sided regular polygon is 360°, Find the measure of each angle. Number of sides = n = 30

∴ Measure of each exterior angle of a regular polygon = Do It Together

(2x − 1)°

(8x + 3)°

∠A = (2x - 1)° = 23°; ∠B = (3x - 5)° = 31°;

Example 6

Find the measure of the missing angles a, b, c and d. a 34°

122°

34°

71°

360º 360º = = 12º. n 30

Error Alert! The sum of an interior angle and the adjacent exterior angle can never be greater than 180°.

b 81°

∠a + 122° = 180°. So, ∠a = 180° - 122° = 58°

115°

90°

∠a = 58°; ∠b = _____, ∠c = _____, ∠d = _____ Chapter 3 • Polygons

Do It Yourself 3C 1 Fill in the blanks. a The polygon in which the sum of all exterior angles is equal to the sum of interior angles is called ______. b The sum of interior angles of a polygon of n sides is ______ right angles. c The measure of each exterior angle of a regular polygon of 18 sides is ______. d In case of a regular polygon that has 8 sides, each interior angle is ______ and each exterior angle is ______.

2 Write if true or false. a The sum of all exterior angles of a polygon is always 360°. b It is possible to have a polygon whose exterior angle is 175°. c It is possible to have a regular polygon whose each exterior angle is d Each exterior angle = (n - 2) ×

180º . n

1 8

th of a right angle.

3 Find the measure of each exterior angle of the following regular polygons with: a 3 sides

b 5 sides

c 8 sides

d 12 sides

e 24 sides

4 Find the number of sides of the regular polygons if each of their exterior angles are: a 40

b 45

c 60

d 72

e 90

5 The measure of the exterior angle of a hexagon is (3x – 4)°, (x + 4)°, (7x – 3)°, (8x – 1)°, (2x + 3)°, (9x + 1)°. Find the measure of each angle.

Word Problem 1

A carpenter is building a new table. The table is in the shape of a regular heptagon. What is the measure of each interior angle of the tabletop.

Points to Remember • A polygon is a simple closed curve made up of straight line segments, with no overlapping or self-intersecting parts. • The polygons can be classified into different types on the basis of their number of sides and vertices.

• A diagonal of a polygon is a line segment that joins two vertices of the polygon, which are not already joined by the adjacent vertices of the polygon.

• In a convex polygon, all the diagonals lie inside the polygon, whereas in a concave polygon, at least one diagonal lies outside the polygon. • • • • 40

The number of diagonals in a polygon with n number of sides = n(n - 3) ÷ 2

If a polygon n has number of sides, then the sum of all the interior angles is (n - 2) × 180° The sum of exterior angles in any polygon is always a constant, i.e. 360°.

If a polygon has n sides, the measure of each exterior angle of a regular polygon = 360° ÷ n.

Math Lab Exploring Polygons Setting: Individual Materials: Pen and paper Method:

Each student will explore the classroom and write the names of 5 polygons they spot.

They will then identify the polygon as concave or convex.

The student who completes the task first, wins.

Chapter Checkup 1 Fill in the blanks. a A polygon with all sides and angles equal is called ________. b The sum of the interior angles of a hexagon is ________. c A polygon that has at least one interior angle more than 180° is called: d The number of diagonals in a pentagon is ________. e The sum of the exterior angles of any polygon is always ________.

2 How many diagonals will each of these polygons have? a Convex quadrilateral

b Triangle

c Polygon with 17 sides

3 Find the number of sides of the polygon that has the number of diagonals given: a 27

b 44

c 55

d 62

e 67

4 Identify the polygons as concave or convex. a

5 Find the interior angle sum of the polygons with sides: a 9

b 13

c 16

d 21

e 33

d 3780

e 4500

6 Find the number of sides whose interior angle sum is: a 1800

b 2700

c 3240

7 Find each exterior angle of the regular polygons with the number of sides. a 9

b 15

c 20

d 36

e 45

d 90°

e 120°

8 Find the number of sides of a regular whose exterior angle is: a 40° Chapter 3 • Polygons

b 45°

c 72°

9 If two adjacent angles of a parallelogram are (5x - 5)° and (10x + 35)° then find the ratio of these angles. 10 The angles of a pentagon are x°, (x - 5)°, (x + 15)°, (3x - 44)° and (x - 70)°. Find x. 11 How many non-overlapping triangles can we make in a polygon having n sides by joining the vertices? 12 The angles of a quadrilateral are in the ratio 1:2:3:4. Find the difference between the largest and the smallest angle.

13 The angles of a hexagon are in the ratio 1:2:3:4:6:8. Find the measure of the smallest and the biggest angles. 14 The measure of four angles of a heptagon is equal and the measure of the other three angles is 120° each. Find the measure of the unknown angles.

15 Is it possible to have a regular polygon of which the exterior angle is 50°? 16 Each interior angle of a polygon is 5 times the exterior angle of the polygon. Find the number of sides. 17 Find the total number of diagonals if the interior angle sum of a polygon is 1800°. 18 The sum of all the interior angles of a polygon is 4 times the sum of its exterior angles. Find the number of sides in the polygon. Also, find the measure of each exterior angle and each interior angle.

19 Find the minimum interior angles and maximum exterior angles possible in a regular polygon. 20 The ratio between an exterior angle and the interior angle of a regular polygon is 1:2. Find the: a measure of each exterior angle

b measure of each interior angle.

c number of sides in the polygon.

Word Problems

spider builds its web in the shape of a regular hexagon. How many diagonals does the A spider have in its web? If it builds a web with an octagon, how many diagonals would be there?

arika drew a polygon where all sides and angles were equal. Is the polygon regular or S irregular?

ohan is designing a logo for his company which is in the shape of a regular 12-sided S polygon. How many diagonals connect the interior angles of the polygon? How many triangles can be formed by connecting these interior angles?

ama works in a factory that produces footballs. She needs to calculate the angle measure R of the shapes on the soccer ball. What is the measure of each interior angle of the regular pentagons?

akshi drew a triangle in which the second angle is double the first angle and the third angle S is 40°. Find all three angles of the triangle that Sakshi made.

In a city with several towns connected by bridges, there are 10 towns. If each town is connected to every other town by a direct bridge, how many bridges are needed in total? How many diagonals does this network of bridges have?

4 Quadrilaterals

Let's Recall

Quadrilaterals are geometric shapes with 4 sides, 4 corners or vertices, and 4 angles. They can be seen in books, roofs of buildings, earrings, etc.

Quadrilaterals come in various forms, each with its unique properties. To understand quadrilaterals better, let's explore some key terms. Diagonals

Opposite Angles

Adjacent Sides

These are the line segments joining opposite vertices of a quadrilateral.

These are the angles in a quadrilateral with no common arm.

E.g. – RT, US

E.g. – (∠T, ∠R ), (∠U, ∠S)

Adjacent sides of a quadrilateral are sides that share a common vertex (corner). They are next to each other in the sequence of sides.

Opposite Sides Opposite sides of a quadrilateral are sides that do not share a common vertex. They are located on opposite sides of the quadrilateral. E.g. – (RS, UT), (ST, RU)

E.g. – (RU, UT), (SR, ST), etc. Adjacent Angles S

Adjacent angles of a quadrilateral are angles that share a common side. E.g. – (∠R, ∠U ), (∠U, ∠T), (∠T, ∠S), (∠S, ∠R)

Letʼs Warm-up Match the columns based on the quadrilateral PQRS given below. Column A

Column B

1 Vertices

PS, QR

2 Sides

∠Q, ∠S

3 Adjacent Sides

P, Q, R, S

4 Opposite sides

PS, SR

5 Opposite Angles

PS, SR, QR, PQ

P S

I scored _________ out of 5.

Understanding Quadrilaterals and their Properties Real Life Connect

Once upon a time in a quiet neighbourhood, Max lived with his uncle, Charlie. One day, they found a colourful kite soaring in the park. Charlie explained that the kite was shaped like a quadrilateral. The kite’s tail was a rectangle, a type of quadrilateral. Max was thrilled. He realised that shapes were all around him, even in a quiet neighbourhood. Max couldn’t wait for more adventures with Uncle Charlie, discovering quadrilaterals everywhere they went.

Classifying Quadrilaterals There are different kinds of quadrilaterals depending upon their properties, like the kite being flown by Max that was a different quadrilateral than the tail of the kite which was a rectangular strip. Let’s explore some more information about quadrilaterals.

Properties of Quadrilaterals In this section, we will see some properties of quadrilaterals and understand regions in and around quadrilaterals.

Interior of a Quadrilateral The interior of a quadrilateral refers to the region enclosed by the four sides of the quadrilateral.

Exterior of a quadrilateral Interior of a quadrilateral

Exterior of a quadrilateral

Exterior of a Quadrilateral The exterior of a quadrilateral refers to the region outside the four sides of the quadrilateral.

Quadrilaterals can be convex or concave B

Convex Quadrilaterals In these quadrilaterals measure of each angle is less than 180°. D

C Q

Concave Quadrilaterals R P

In these quadrilaterals the measure of one of the angles is more than 180°. Here ∠R is greater than 180°. S

Let’s say we have 2 quadrilaterals with the interior angle measures as given. Identify the type of each quadrilateral. 1 Quadrilateral ABCD with ∠A = 45°, ∠B = 105°, ∠C = 95°, ∠D = 115°. Since none of the angles measures more than 180°, ABCD is a convex quadrilateral. 2 Quadrilateral PQRS with ∠P = 34°, ∠Q = 201°, ∠R = 71°, ∠S = 54°. ∠Q = 201° which is greater than 180°. So, PQRS is a concave quadrilateral. Example 1

Among the following angles, which one cannot be an interior angle in a convex quadrilateral? 1 73°

2 85°

3 173°

4 211°

In a convex quadrilateral, no angle can exceed 180°. Therefore, the 211° angle cannot be one of the angles in a convex quadrilateral. Example 2

Which of the following angles can be one of the interior angles in a concave quadrilateral? 1 33°

2 119°

3 183°

4 Any of the 3 angles given in a, b and c

In a concave quadrilateral, one of the angles can be more than 180°. Therefore, the correct option is d since any of the 3 angles can be interior angles in a concave quadrilateral. Example 3

Place the points J, K, L, M, N and O given below in the quadrilaterals so that: Only points J, L and M are in the interior.

Only points K, L and O are in the interior.

Only points L, M and N are in the interior.

J L

N Do It Together

Classify the following as convex or concave quadrilaterals based on the angles of the quadrilaterals. Angles

150o, 100o, 40o, 70o

Type of Quadrilateral

75o, 105o, 101o, 79o

216o, 36o, 40o, 68o

Convex

Trapezium and Kite Trapezium: A quadrilateral with exactly one pair of opposite sides parallel is a trapezium. Let us see trapezium shapes in real life.

Did You Know? The word “trapezium” comes from the Greek “trapezion,” meaning “a little table.” It was originally used to describe a quadrilateral with no parallel sides.

Chapter 4 • Quadrilaterals

Properties of trapezium

Given below are the properties of a trapezium. • • •

One pair of opposite sides is parallel and the other pair of opposite sides is non-parallel. Here, AB ǁ DC When the non-parallel sides in a trapezium are equal, it is known as an isosceles trapezium. PS = QR. The diagonals in an isosceles trapezium are equal. PR = QS. A

Supplementary angles in trapezium ABCD. ∠A + ∠D = 180° D

∠B + ∠C = 180°

Kite A kite is a quadrilateral with two pairs of consecutive sides of equal length.

Think and Tell How many pairs of opposite angles in a kite are equal?

Example 4

•

AC ⊥ BD

•

Diagonal BD bisects the angle ∠B and ∠D.

•

∠A = ∠C.

•

Only diagonal BD bisects the diagonal AC and not vice-versa.

•

Diagonal BD divides the kite into two congruent halves.

Look at the kite. Find angles ∠CAD, ∠ADC and ∠ABC. B

In ΔABC, ∠BAC = ∠BCA = 50° (Angles opposite to equal sides) ∠BAC + ∠BAC + ∠ABC = 180° (Angle sum property of the triangle)

50° + 50° + ∠ABC = 180° ⇒ ∠ABC = 180° - 100° = 80°

50°

Also, ∠ADB = ∠CDB = 27° (BD bisects ∠D) Hence, ∠ADC = ∠ADB + ∠CDB = 27° + 27° = 54°

27°

In Δ ADC, ∠ACD = ∠CAD (angles opposite to equal sides) ∠ACD +∠CAD +∠ADC = 180° (Angle sum property of the triangle) 2∠CAD + 54° = 180°

2∠CAD = 180° - 54° = 126°. So, ∠CAD = 63°

Example 5

PQRS is a trapezium where PQ ǁ SR. Find the values of x and y. P

S Do It Together

107°

126°

(2y + 3)°

(4x − 2)°

From the figure,

Similarly,

∠P + ∠S = 180° (supplementary angles)

∠Q + ∠R = 180° (supplementary angles)

2y + 3 + 107 = 180

4x - 2 + 126 = 180

2y = 180 - 110 = 70 70 y= = 35 2

4x = 180 - 124 = 56 56 x= = 14 4

Find the measure of ∠A and ∠C in the figure given below. B A

C 54° D

Parallelograms Parallelogram: A parallelogram is a quadrilateral with opposite sides that are both parallel and equal in length. We see parallelograms in roofs, solar panels, and many other places. Can you think of 2 other places where you see parallelograms? Properties of a parallelogram •

Opposite angles of a parallelogram are equal. Here, ∠P = ∠R;

∠S = ∠Q. • Opposite sides of a parallelogram are parallel and equal. Here, PQ = SR, PS = QR; PQ // SR; PS // QR. • Diagonals of a parallelogram bisect each other. Here, the diagonals PR and SQ are bisecting at point O.

Let us see some proofs of parallelograms 1 I f a pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram. Let PQRS be a quadrilateral such that PQ = SR and PQ ǁ SR. Consider ΔPQR and ΔRSP, then PQ = SR

(Given)

∠QPR = ∠SRP

(Alternate angles as PQ ǁ RS)

PR = RP Hence, ΔPQR ≅ ΔRSP

(Common side)

(SAS Congruency rule)

Therefore, ∠PRQ = ∠RPS (Corresponding parts of congruent triangles) Since ∠PRQ and ∠RPS are alternate angles, PS ǁ QR

As the opposite sides have been shown to be parallel, we can conclude that PQRS is a parallelogram. Chapter 4 • Quadrilaterals

2 Any two adjacent angles of a parallelogram are supplementary. Let PQRS be a parallelogram. Consider PS ǁ QR and PQ as their transversal. Then, ∠SPQ and ∠RQP are supplementary as they are the interior angles on the same side of the transversal PQ.

Similarly, considering PS ǁ QR and SR as their transversal. ∠PSR and ∠QRS are supplementary as they are the interior angles on the same side of the transversal SR.

Extending the same reasoning it can be shown that ∠QPS & ∠RSP and ∠SRQ & ∠PQR are also supplementary considering PQ ǁ RS along with PS and QR as its transversals respectively. 3 Diagonals of a parallelogram bisect each other. Let PQRS be a parallelogram whose diagonals QS and PR intersect at T. Consider ΔPTQ and ΔRTS, then PQ = SR

(Opposite sides of a parallelogram)

∠TQP = ∠TSR

(Alternate angles as PQ ǁ RS)

∠PTQ = ∠RTS

(Vertically opposite angles)

Hence, ΔPTQ ≅ ΔRTS

(AAS criteria of congruency)

Therefore, PT = TR and QT = ST (Corresponding parts of congruent triangles) 4 Opposite sides and angles of the parallelogram are equal. Let PQRS be a parallelogram with diagonal PR shown in the fig.

Consider ΔPQR and ΔRSP, then

a PR = RP (common side) b ∠QPR = ∠SRP

(Alternate angles as PQ ǁ RS)

(Alternate angles as PS ǁ QR)

∠QRP = ∠SPR

Hence, ΔPQR ≅ ΔRSP

(ASA axiom of congruency)

Therefore, ∠Q = ∠S, QR = PS and PQ = SR (Corresponding parts of congruent triangles) From (b) and (c), ∠QPR + ∠SPR = ∠SRP + ∠QRP ⇒ ∠P = ∠Q. Therefore, all the opposite sides and angles have been proven to be equal.

Rhombus, Square and Rectangle Rectangle

A rectangle is a quadrilateral with four right angles and opposite sides of equal length.

Error Alert! Diagonals of a parallelograms may or may not bisect angles through which they pass.

Properties of rectangle •

All the angles in a rectangle are right angles and equal.

Here, ∠A = ∠B = ∠C = ∠D = 90°

• • •

Opposite sides of a rectangle are parallel and equal. Here, AB ǁ DC; AD ǁ BC and AB = DC; AD = BC

The diagonals are equal and bisect each other.

Rhombus A rhombus is a four-sided geometric shape with all sides of equal length and opposite angles of equal measure, but the angles are not necessarily right angles.

Properties of Rhombus: • • • •

All the sides of a rhombus are equal. Here, PQ = PS = QR = SR

Q O

Opposite sides of a rhombus are parallel. Here, PQ ǁ SR; PS ǁ QR

Diagonals bisect each other at right angles. Here, PQ = RO; OQ = OS Opposite angles of a rhombus are equal. Here, ∠P = ∠R; ∠S = ∠Q

Square

A square is a quadrilateral with equal-length sides and four right angles.

Properties of Square: • • • •

Example 6

All the sides of a square are equal. Here, AB = BC = CD = DA

B O

All the angles in a square are right angles. ∠A = ∠B = ∠C = ∠D = 90° Opposite sides are parallel. Here, AB ǁ CD; AD ǁ BC.

iagonals are equal and bisect each other at right angles. AO = CO; D OD = OB.

Given that ABCD is a parallelogram. Find the values of x and y. We know that opposite sides in a parallelogram are equal. Hence, 2x + 3 = 35 cm

4y + 7 = 23 cm

2x = 35 - 3 = 32 32 x= = 16 cm 2

4 y = 23 – 7 = 16 16 y= = 4 cm 4

Chapter 4 • Quadrilaterals

35 cm

D 23 cm

4y + 7 B

2x + 3

Example 7

If the diagonals of a rhombus measure 10 cm and 24 cm, determine the length of one of its sides. Let PQRS be a rhombus where diagonals PR = 24 cm and QS = 10 cm. Since the diagonals of a rhombus bisect each other at right angles OQ = 5 cm and OP = 12 cm. P

In ΔPOQ, we have OQ2 + OP2 = PQ2 (Pythagoras' Theorem) 52 + 122 = PQ2

25 + 144 = PQ2 PQ2 = 169. So, PQ = 13 cm Example 8

The diagonals of rectangle PQRS intersect at T. If ∠QTR = 44°, find ∠TPS.

R P

∠QTR = ∠PTS = 44° (vertically opposite angles) Since, the diagonals of a rectangle are equal and they bisect each other, PT = ST ⇒ ∠TPS = ∠TSP (Angles opposite to equal sides)

44°

∠TPS + ∠TSP + ∠PTS = 180° (Angle sum property of triangle) 2∠TPS + 44° = 180° 2∠TPS = 136° ∠TPS = Example 9

136° = 68° 2

If one of the sides of a parallelogram is 15 mm and its perimeter is 80 mm, find the length of the other sides. Since opposite sides in a parallelogram are equal, let the sides be 15 mm, x, 15 mm and x. 15 mm + x + 15 mm + x = 80 mm (Given) 2x + 30 mm = 80 mm 2x = 80 mm – 30 mm 2x = 50 mm or x = 25 mm Hence, the length of sides in the parallelogram are 15 mm, 25 mm, 15 mm and 25 mm.

Example 10

Given that JKLM is a rhombus. Find the values of x॰, y॰ and z॰. In ΔJKM, JM = JK (Sides of a rhombus) Therefore, ∠JMK= ∠JKM = x Also, 120° = ∠JMK + ∠JKM (Exterior angle is the sum of 2 interior angles) x + x =120° or 2x = 120° ⇒ x = 60°

120° x°

Also, x = y = 60° (Alternate angles)

In ΔTLM, y° + z° + ∠LTM = 180° (Angles sum property of triangle) Substituting the values, 60°+ z°+ 90° = 180° z° + 150° = 180° ⇒ z° = 180° - 150° = 30°

y°

z°

Example 11

Find the values of a and b if PQRS is a parallelogram. We know that diagonals in a parallelogram bisect each other. Therefore P

a−

a + 2b = 18 ⇒ 8 + 2b = 18

10 ⇒ 2b = 18 - 8 = 10 ⇒ b = =5 2 Hence, the values of a and b are 8 and 5 respectively. Do It Together

a-3=5⇒a=3+5=8

a O +2 b R

ABCD is a parallelogram and PQRS is a square. Find the value of the unknown angles in the given figures. D

110°

Do It Yourself 4A 1 Place the points A, B, C, D, E and F given below in the quadrilaterals such that:

a Only points A, B and C are in the interior.

b Only points B, C and D are in the interior.

B F

c Only points B, F and E are in the interior.

2 Classify the following as convex or concave quadrilateral based on the angles of the quadrilateral. a 190o, 60o, 30o, 80o

b 65o, 115o, 111o, 69o

3 Given that JKLM is a rectangle. Find the value of ∠MJL and ∠TJK. J

c 26o, 226o, 30o, 78o

4 Find the value of x and y in the figure shown below.

P 119°

T 140° M

5 Use the properties of a kite to find the value of ∠QPT and ∠TRS.

(2y + 7)°

(5x − 2)°

quadrilateral a parallelogram?

)

-6

(2a

(5a 11)

)

-1

(1 + 2b)

50°

(3b

6 What value of ‘b’ would make the given

61° T

133°

Chapter 4 • Quadrilaterals

7 PQRS is a parallelogram in which one angle measures 80°. Determine the measures of the remaining three angles in the parallelogram.

8 The length of a parallelogram is 15 cm longer than its breadth. If the perimeter of the parallelogram is 130 cm, find the length and breadth of the parallelogram.

9 WXYZ is a parallelogram where two adjacent angles have a ratio of 2:5. Calculate the measures of all the angles in the parallelogram.

10 PQRS is a rhombus whose diagonals intersect at Z. Draw a rough diagram and answer the questions given below. a Name any four pairs of line segments that equal in length.

b Name any four pairs of angles that equal in measure.

c Identify three pair of congruent triangles in this rhombus.

d Is ∠QRZ = ∠ZRS? e Is ΔQPZ ≅ ΔSPZ?

Word Problem 1

Emily is walking through a square park with sides of length 200 metres. What is the shortest distance she can travel to reach the opposite corner?

Angle Sum Property of Quadrilaterals Interior Angle Sum Property of Quadrilateral The sum of interior angles of a quadrilateral are 360°. Let us prove using figures. Consider the quadrilateral ABCD. Notice the measure of the angles ∠A, ∠B, ∠C and ∠D given below. If we cut each of the interior angles of the quadrilateral and place them side by side, we will get a complete angle i.e. 360°. This is true for any quadrilateral. The angles will always add up to 360°. Hence, we can conclude C that the sum of angles of a quadrilateral is 360°.

100°

B 115°

75°

Exterior Angle Sum Property of Quadrilateral

The angles that are formed between one side of a quadrilateral and another line extended from an adjacent side are called its exterior angles. Here, ∠w, ∠x, ∠y and ∠z are the interior angles. ∠1, ∠2, ∠3 and ∠4 are the exterior angles.

70°

100°

115°

75°

∠4

P ∠1 ∠w

∠X ∠Z ∠2

∠Y

∠3

The sum of the measures of the exterior angles of a quadrilateral is 360°. Hence, ∠1 + ∠2 + ∠3 + ∠4 = 360°. Linear pair of exterior and interior angles are as follows: ∠1 and ∠w, ∠2 and ∠z, ∠3 and ∠y, ∠4 and ∠x ∠1 = 120°, ∠2 = 60°, ∠3 = 65° and ∠4 = 120° 120°, + 60° + 60° + 20° = 360° So, ∠1 + ∠2 + ∠3 + ∠4 = 360° 52

Remember! Every interior angle in a convex quadrilateral has its corresponding exterior angle.

Example 12

Find the value of x° in the diagram given below.

Example 13

ind the value of y° in the diagram given F below. 109°

x°

85°

y°

71°

79° 114°

y°

Using the interior angle sum property of the quadrilateral,

U sing the exterior angle sum property of the quadrilateral,

x° + 278° = 360°

2 y° + 180° = 360°

x° + 85° + 79° + 114° = 360°

y ° + y° + 71° + 109° = 360°

x° = 360° - 278° = 82°

2 y° = 360° - 180° = 180° 180° y ° = = 90° 2

If one of the angles in a parallelogram is 65°, find the measure of other angles.

Example 14

Since opposite angles in a parallelogram are equal, let the angles be75°, x, 75° and x. 65° + 65° + x + x = 360° (Angle sum property of Quadrilateral) 130° + 2x = 360° 2x = 360° - 130° = 230 ⇒ x = 230 ÷ 2 = 115°. Hence, the angles of the parallelogram are 65°, 115°, 65° and 115°. Do It Together

Match the set of interior angles of quadrilaterals in the left column with the correct missing angle in right column. Interior Angles of a Quadrilateral

Missing Angle

1 55°, 75°,________, 145°

80°

2 35°, ________ , 110°, 140°

85°

3 60°, 100°, 120°, ________

40°

4 ________ , 90°, 70°, 160°

75°

Do It Yourself 4B 1 Which of the set of angles given below is NOT a set of interior angles of a quadrilateral? a 47°, 58°, 98°, 157°

b 54°, 59°,72°,108°

c 63°, 107°, 74°, 116°

d 83°, 68°, 82°, 127°

2 Which of the set of angles given below is NOT a set of exterior angles of a quadrilateral? a 57°, 82°, 98°, 123°

b 54°, 59°, 72°, 108°

c 65°, 85°, 85°, 125°

d 49°, 78°, 103°, 130°

Chapter 4 • Quadrilaterals

3 Find the value of the unknown angle in each of the following. a

75°

23° 20°

100°

b° 93°

71° a°

110°

22°

e°

68°

79° c°

81°

d°

51°

15°

f°

54°

28°

138°

58°

47°

4 What could be the value of the unknown exterior angle in each case? a

76°

y°

103°

z°

111°

x°

131°

46°

78°

80°

103° d

86°

126°

51° 100°

t°

u°

53°

u°

111°

5 Find the value of unknown angles in each of the cases given below. a

83° x°

(2x + 10)°

37°

94°

z°

4y°

(3y − 30)°

y°

27°

x° z°

68°

6 In a parallelogram, two adjacent angles are (3y - 10)° and (2y + 15)°. Find the measures of all four angles of the parallelogram.

7 In a parallelogram, one angle measures 60° less than its adjacent angle. Determine the measures of all four angles in the parallelogram.

8 RSTU is a parallelogram in which the sum of two opposite angles is 150°. Find the measures of all four angles in the parallelogram.

9 Find the values of all the angles in the quadrilateral given below

10 Find the values of unknown angles such that ABCD is a parallelogram. A z°

(7x − 3)° (2x + 10)°

(3x + 4)° (5x + 9)°

73°

x°

y° 45°

Word Problem 1

Rahul pins four nails to a soft wooden board. He then proceeds to tie threads between

these nails so that they form a quadrilateral. The total length of thread he used was 12 cm. The angles of this quadrilateral are in the ratio 1:2:3:4. Is the quadrilateral formed by the threads a convex or a concave quadrilateral?

Points to Remember •

A quadrilateral has four sides and four vertices.

• The sum of interior angles of a quadrilateral is 360°. The sum of exterior angles of a quadrilateral is 360°. •

Opposite sides of the parallelogram are parallel and equal and the diagonals bisect each other.

• Opposite sides and angles of the parallelogram are equal and any two adjacent angles of a parallelogram are supplementary. • In a trapezium, one pair of opposite sides is parallel while the other pair of sides in non-parallel. •

Adjacent pair of sides are equal in a kite.

Math Lab Setting: In groups of 3-4.

Quadrilateral Sorting Challenge

Materials Required: Quadrilateral cards (pre-made or printed), large poster paper or whiteboard, markers, scissors, tape Method:

1 Discuss the concept of quadrilaterals like rectangle, square, rhombus, trapezium, parallelograms, etc. 2

Create cards with quadrilateral names and key properties.

Chapter 4 • Quadrilaterals

Draw a table with four columns, label it with quadrilateral names, and tape down cards.

Divide the class into groups and provide cards and sorting table access.

5 Instruct groups to sort cards into correct columns based on properties, encouraging discussions. 6

Gather the class to discuss choices and have groups explain their reasoning.

Chapter Checkup 1 State whether the following statements are true or false. a The diagonals of a kite bisect each other at right angles. b The diagonals of an isosceles trapezium are equal. c The opposite angles of a trapezium may be equal. d The diagonals of a rectangle are perpendicular to each other. e All the quadrilaterals are parallelograms.

2 Which of the interior angles given below would not belong to a convex quadrilateral? b 178°

a 100°

d 2°

c 222°

3 Classify the points lying in the interior and exterior of the quadrilateral. P R

4 Which of the sets of angles given below is a set of interior angles of a quadrilateral? a 47°, 58°, 98°, 156°

b 54°, 59°, 72°, 108°

c 63°, 100°, 74°, 116°

d 83°, 68°, 82°, 127°

5 Which of the sets of angles given below is a set of exterior angles of a quadrilateral? a 57°, 82°, 98°, 120°

b 53°, 69°, 77°, 111°

c 65°, 85°, 85°, 125°

d 49°, 75°, 103°, 130°

6 Write the set of exterior angles corresponding to the set of interior angles given below. a 70°, 90°, 110°, 90°

b 45°, 60°, 100°, 155°

c 80°, 85°, 95°, 100°

7 Find the value of unknown interior angle in each case. 41°

88°

97°

m°

l° 21°

58°

64°

58°

66°

n° 74°

8 Find the value of unknown exterior angle in each case. a

p° 117°

48°

93°

r°

93°

89°

92°

67°

q°

9 In a parallelogram, the adjacent angles are (6z - 25)º and (3z + 10)º. Determine the measures of all four angles in the parallelogram.

10 The length of a parallelogram is 20 cm longer than its breadth. If the perimeter of the parallelogram is 140 cm, find the length and breadth of the parallelogram.

11 WXYZ is a parallelogram in which one angle is 140º. Find the measures of the other three angles in the parallelogram.

12 In a parallelogram, two adjacent angles are in the ratio 3:4. Determine the measures of all four angles in the parallelogram.

13 In a rhombus, one of the diagonals is of the same length as one of its sides. Determine the measures of the angles in the rhombus.

14 Find the values of wº, xº, yº and zº. a

125° w°

50°

x° y°

z° 135°

135°

125°

3y + 5

6y + 4 C

15 Prove that the diagonals of a rhombus bisect its angles.

79°

C Q

16 Find the measure of all the interior angles of the rhombus PQRS.

T S

52°

Word Problems 1

At the local playground, there is a climbing structure made of four bars forming a

quadrilateral shape. The angles at the corners are measured and found to be in the ratio 2:3:4:5. Determine whether the climbing structure is a parallelogram.

A tennis court is designed with four sides that form a parallelogram shape. The length of

one side is 24 m, and the angle at the top corner measures 120°. What is the length of the opposite side, and what is the measure of the angles at the other corners?

Chapter 4 • Quadrilaterals

Construction of

Quadrilaterals

Let's Recall Rhea drew a figure with different quadrilaterals. She asked her friend Gyan to guess the shape. Gyan was confused since all of them had 4 sides, 4 angles and 4 vertices. Let us identify the shapes using their properties. Properties of some special quadrilaterals A

Kite

Trapezium

Rhombus Parallelogram

Parallelogram

•

Opposite sides are equal and parallel

• Opposite angles are equal, and diagonals bisect each other

E D

C B

• All sides are equal

•

Rhombus

Opposite sides are parallel Opposite angles are equal

Diagonals bisect each other at right angles Kite

• Two pairs of adjacent sides are equal

B C

A D D

• Diagonals intersect at right angles

Trapezium • One pair of sides is parallel, and the other pair is non-parallel

• Diagonals in an isosceles trapezium are equal A

Letʼs Warm-up

Name the quadrilateral by reading the clues. 1 Exactly two pairs of adjacent sides are equal. _________________ 2 Exactly one pair of opposite sides are parallel. _________________ 3 All angles in a quadrilateral are right angles but only opposite sides are equal. _________________ 4 All sides are equal but all angles may not necessarily be equal. _________________ I scored _________ out of 4.

Real Life Connect

Rahul looked at building sketches and wondered how his architect aunt, Nilima, designed such amazing structures.

Nilima told him that she uses rulers and protractors to create shapes, which amazed Rahul. He was particularly eager to learn about making shapes, especially quadrilaterals.

Constructing Quadrilaterals To create quadrilaterals, we need a minimum number of elements. For instance, for a unique triangle, we need three measurements (at least one being a side). A quadrilateral requires at least five of its ten elements (four sides, four angles and two diagonals) to be given for a unique construction. You can construct a quadrilateral with the following sets of at least five elements. 1

Four sides and one diagonal

Three sides and two diagonals

Three angles and two included sides

mathematician, Brahmagupta,

Three sides and two included angles

quadrilaterals in his book

Four sides and one angle

Did You Know? In the 7th century, Indian explored properties of unique “Brāhma-sphuṭa-siddhānta.”

Let us discuss constructing quadrilaterals in each of the cases listed above one by one.

Construction with 4 Sides and 1 Diagonal A quadrilateral can be constructed when any 4 sides and one diagonal are given.

1 D raw a line segment PR = 6.5 cm.

2 D raw two arcs: one with center at P (5.3 cm radius) and another with center at R (4.2 cm radius), intersecting at Q.

3.7 cm

Draw a rough sketch of the quadrilateral PQRS with the marked measurements marked. We may divide quadrilateral PQRS into two triangles namely RSP and PRQ.

cm 5.

6.5 cm

5.3 cm

4.2

6.5 cm

3 Join PQ and QR.

4.5 cm

6.5 cm

4.2 cm

Let us construct a quadrilateral PQRS in which PQ = 5.3 cm, QR = 4.2 cm, RS = 4.5 cm, SP = 3.7 cm and PR = 6.5 cm.

4 D raw two arcs on the opposite side of PR: one with center at P (3.7 cm radius) and another with center at R (4.5 cm radius), intersecting at S.

5 Join PS and RS.

5. 3

cm 5. 3 P

S Example 1

6.5 cm

4.2

6.5 cm

m N

Construct a quadrilateral LMNO in which LM = 6.8 cm, MN = 4.5 cm, NO = 5.2 cm, OL = 3.9 cm and LN = 7.1 cm.

Draw a sketch of a quadrilateral LMNO with measurements roughly marked. We may divide the quadrilateral LMNO into two triangles namely ∆LMN and ∆LNO.

3.9 cm

7.1 cm

6.8 cm M

1 Draw a line segment LN = 7.1 cm.

4.5

2 D raw two arcs: one with center at L (6.8 cm radius) and another with center at N (4.5 cm radius), intersecting at M. 3 Join LM and MN.

9 cm

5 Join OL and ON. LMNO is the required quadrilateral.

86° 7.1 cm

4 D raw two more arcs: one with center at L (3.9 cm radius) and another with center at N (5.2 cm radius), intersecting at O.

Do It Together

4.5 cm

c O 5.2

Construct a quadrilateral FEDC with given measures. Complete the rough figure. CD = 4.3 cm, DE = 3.6 cm, EF = 5.6 cm, FC= 4.7 cm, CE = 6 cm

Construction with 3 Sides and 2 Diagonals

S 3.5 cm

A quadrilateral can be constructed when any three sides and two diagonals are given.

1 Draw a line segment PQ = 5.5 cm.

2 Draw two arcs: one with center

at P (4.9 cm radius) and another with center at Q (6.7 cm radius), intersecting at S.

4.9 c P

S 4.9 cm

5.5 cm

Chapter 5 • Construction of Quadrilaterals

5.5 cm

3 Join PS and QS.

Draw a sketch of the quadrilateral PQRS with the measurements roughly marked.

7 6.

Let us construct a quadrilateral PQRS in which PQ = 5.5 cm, PS = 4.9 cm, RS = 3.5 cm, diagonal PR = 6.1 cm and diagonal QS = 6.7 cm.

5.5 cm

4 Draw two arcs: one with center at P (6.1 cm radius) and another with center at S (3.5 cm radius), intersecting at R.

4.9 cm

5.5 cm

3.5 cm

Example 2

4.9 cm

5 Join RS and QR. PQRS is the required quadrilateral.

5.5 cm

Construct a quadrilateral DAVE in which DA = 3.6 cm, DE = 5 cm, EV = 3 cm, diagonal AE = 6.2 cm and diagonal DV = 5.8 cm. Find the length of side AV.

A 5.1 cm

5.1 cm 6.2

3.6 cm

5. D

Do It Together

3 cm

cm 5 cm

5 cm

1 Draw a line segment DE = 5 cm. 2 D raw two arcs: one with center at D (5.8 cm radius) and another with center at E (3 cm radius), intersecting at V. 3 Join DV and EV.

4 D raw two arcs: one with center at D (3.6 cm radius) and another with center at E (6.2 cm radius), intersecting at A. 5 J oin DA and AE. DAVE is the required quadrilateral. On measuring, we get the length of AV as 5.1 cm.

Construct a quadrilateral LISA in which LI = 4.2 cm, SI = 5.5 cm, SA = 3.8 cm, diagonal IA = 6.7 cm and diagonal LS = 7.6 cm. Determine the length of the side LA.

Construction with 3 Angles and 2 Included Sides

120°

Let us construct a quadrilateral ABCD in which AB = 4.9 cm, BC = 3 cm, ∠A = 60°, ∠B = 90° and ∠C = 120°. Draw a rough sketch of the quadrilateral ABCD and mark its measurements.

3 cm

A quadrilateral can be constructed when any three angles and 2 included sides are given.

3 cm

Draw a sketch of the quadrilateral DAVE with the measurements roughly marked.

2 6.

3.6 cm

60° A

90°

4.9 cm

1 D raw a line segment AB = 4.9 cm.

2 Construct ∠XAB = 60° at A.

3 Construct ∠YBA = 90° at B. X

90°

60° B

4.9 cm

4 U sing B as centre and a radius of 3 cm, draw an arc cutting BY at C. X

4.9 cm

60°

4.9 cm

120°

3 cm

90°

4.9 cm

5 A t C, construct ∠ZCB = 120°, so that ZC and XA intersect at D. ABCD is the required quadrilateral.

60°

Y C 3 cm

90° A

60°

4.9 cm

Remember! Keep the compass width consistent when constructing the 90° and 120°.

Construct a quadrilateral PQRS in which PQ = 3.9 cm, QR = 4.5 cm, ∠P = 75°, ∠Q = 100° and ∠R = 60°. Measure the length of PS. Draw a rough sketch of the quadrilateral PQRS with its measurements. 1 Draw a line segment PQ = 3.9 cm.

X Y

R 60°

S 3.3 cm

4.5 cm

75°

3.9 cm

° 00

60°

75° 100°

4.5 cm

Example 3

3.9 cm Q

2 Construct ∠XPQ =75° at A. 3 Draw ∠YQP= 100° at B using a protractor. 4 U sing Q as the centre and a radius of 4.5 cm, draw an arc cutting QY at R. 5 A t R, construct ∠ZRQ = 60°, so that ZR and XP intersect at S. PQRS is the required quadrilateral. On measuring, we get the length of PS as 3.3 cm.

Chapter 5 • Construction of Quadrilaterals

Do It Together

Construct a quadrilateral PUSH in which PU = 4.9 cm, SU = 5.5 cm, ∠P = 75°, ∠U = 100° and ∠S = 60°. Find the sum of the lengths of sides PH and SH.

Construction with 3 Sides and 2 Included Angles

Construct ∠XBC = 60° at B. X

5.4 c

30°

6.6 cm

cm 5.4

60° 6.6 cm

4 Construct ∠YCB = 30° at C. X

6.6 cm

30°

60°

cm 5.4

6.6 cm

6 J oin AD. ABCD is the required quadrilateral.

5.4

cm 5.4

60°

5 U sing C as centre and a radius of 3.3 cm, draw an arc to cut YC at D.

D 30°

6.6 cm

3 U sing B as centre and a radius of 5.4 cm, draw an arc to cut XB at A. A

60°

Draw a rough sketch of the quadrilateral ABCD with its measurements.

3.3

Construct a quadrilateral ABCD in which AB = 5.4 cm, BC = 6.6 cm, CD = 3.3 cm, ∠B = 60° and ∠C =30°.

1 D raw a line segment BC = 6.6 cm.

A quadrilateral can be constructed when any three sides and two included angles are given.

60°

30°

6.6 cm

Construct a quadrilateral PQRS in which PQ = 4.4 cm, QR = 5.7 cm, RS = 2.5 cm, ∠Q = 60° and ∠R = 100°. Find the length of PS.

3 Using Q as centre and a radius of 4.4 cm, draw an arc to cut XQ at P.

4.4 cm Do It Together

100°

5.7 cm

4 Construct ∠YRQ = 100° at R, using the protractor.

2.5 cm

60°

2 Construct ∠XQR = 60° at Q. 4.1 c

60° 100° Q 5.7 cm R

1 Draw a line segment QR = 5.7 cm.

2. 5 c

Draw a rough sketch of the quadrilateral PQRS with its measurements. X

4.4 c m

Example 4

5 U sing R as the centre and having a radius of 2.5 cm, draw an arc to cut YR at S.

6 J oin PS. PQRS is the required quadrilateral. On measuring, we get the length of SP as 4.1 cm.

Construct a quadrilateral JKLM in which JK = 5.7 cm, KL = 4.4 cm, LM = 2.8 cm, ∠K = 60° and ∠L = 120°. Find the measure of ∠J.

Construction with 4 Sides and 1 Angle

cm 2.7

Draw a rough sketch of the quadrilateral PQRS with its measurements. 1 D raw a line segment PQ = 4.7 cm.

2 Construct an angle of 60° at Q. X

60° P

Construct a quadrilateral PQRS in which PQ = 4.7 cm, QR = 3.6 cm, RS = 5.6 cm, SP = 2.7 cm and ∠Q = 60°.

3.6

A quadrilateral can be constructed when four sides and one angle are given.

5.6 cm

4.7 cm

3 W ith the centre Q and radius as equal to 3.6 cm, draw an arc to cut XQ at R. X

R 3.6 cm

4.7 cm

60° Q

Chapter 5 • Construction of Quadrilaterals

4.7 cm

60° P

4.7 cm

4 D raw an arc with center P (2.7 cm radius) and another arc from center R (5.6 cm radius), intersecting the first at S.

5 J oin SP and SR. PQRS is the required quadrilateral.

R 3.6

4.7 cm

5.5 cm

Do It Together

3.5 c

4.7 cm

75° A 4.7 cm B

2 Construct an angle of 75° at B. 3 W ith the centre B and the radius equal to 3.5 cm, draw an arc to cut XB at C.

75°

1 Draw a line segment AB = 4.7 cm.

5 cm

Draw a rough sketch of the quadrilateral PQRS with its measurements.

3.5

Construct a quadrilateral ABCD in which AB = 4.7 cm, BC = 3.5 cm, CD = 5 cm, AD = 5.5 cm and ∠B = 75°. Find the measure of diagonal AC.

5.5

Example 5

4.7 cm

60°

R 3 .6

m 6c

4 D raw an arc with center A (5.5 cm radius) and another arc from center C (5 cm radius), intersecting the first at D. 5 Join AD and CD. ABCD is the required quadrilateral.

On measuring, we get the length of diagonal AC as 5.1 cm.

Construct a quadrilateral EFGH in which EF = 4.7 cm, FG = 3.8 cm, GH = 4 cm, HE = 7 cm and ∠F = 90° Find the sum of the lengths of both diagonals.

Do It Yourself 5A 1 Construct quadrilaterals where the lengths of 4 sides and 1 diagonal are given. a Quadrilateral LMNO: LM = 5.6 cm, MN = 4.2 cm, NO = 7.8 cm, LO = 6.5 cm, MO = 8.1 cm. b Quadrilateral XYZW: XY = 6.3 cm, YZ = 4.7 cm, ZW = 5.2 cm, WX = 7.4 cm, WY = 8.6 cm.

2 Construct quadrilateral GLOW with the given measures. Measure the unknown side. a GL = 5.6 cm, OW = 3.4 cm, GW = 4.8 cm, GO = 6.7 cm, LW = 7.3 cm b GL= 5.2 cm, LO = 4.3 cm, WG = 3.8 cm, GO = 6.1 cm, WL = 7.2 cm

3 Construct the following quadrilaterals when three angles and two sides are given. Find the measure of both of its diagonals.

a ABCD: AB = 7 cm, AD = 4 cm, ∠A = 75°, ∠B = 110° and ∠D = 95°. b WXYZ: WX = 8 cm, XY = 6 cm, ∠W = 120°, ∠X = 85° and ∠Y = 75°.

4 Construct quadrilaterals ABCD with the given measures. a AB = 3.5 cm, BC = 5.4 cm, CA = 4.9 cm, ∠B = 125° and ∠C = 80° b AB = 4.2 cm, BC = 5.3 cm, CA = 3.6 cm, ∠B = 135° and ∠C = 60°

5 Construct the quadrilaterals given below, where 4 sides and one of the angles are known. a RSTU: RS = 2.9 cm, ST = 3.6 cm, TU = 4 cm, RU = 6 cm, ∠S = 90°. b JKLM: JK = 3.9 cm, KL, 3.1 cm, LM = 3.6 cm, MJ = 3.3 cm, ∠ J = 60°.

Word Problem 1

You are creating a poster for your school’s geometry fair and want to include a unique quadrilateral shape. You know that you want to use a quadrilateral with four sides

measuring 9 cm, 6 cm, 12 cm and 7 cm, and you want one of the angles between two

consecutive sides to be 60 degrees. How can you use your ruler and compass to accurately draw this quadrilateral for your poster? Draw two quadrilaterals that fit this description.

Constructing Special Quadrilaterals Some quadrilaterals are special and have some of their measures equal. Some examples of special quadrilaterals are squares, rectangles, rhombus, trapezium, kite and parallelogram. The properties of special quadrilaterals help in constructing them. So, we do not need 5 measures to construct special quadrilaterals. Constructing a Square We know that a square has all its sides equal, all angles measure 90°, its diagonals are equal in length and bisect each other at right angles. A square can be constructed when 1 of its diagonals or 1 of its sides are given.

Chapter 5 • Construction of Quadrilaterals

Let us construct a square with the length of its diagonal as 4.6 cm.

It is known that diagonals of a square are equal in length and bisect each other at right angles. This property can help us construct the square.

Let’s say, the square is PQRS and the diagonals bisect at O.

2.3 cm

Q O 4.6 cm

Given:

PR = QS = 4.6 cm (diagonals are equal)

PO = OR = QO = OS = 2.3 cm (Diagonals bisect each other)

Q 2.3 cm

1 Draw a line segment PR = 4.6 cm. 2 Draw the right bisector XY of PR, intersecting PR at O. 3 With the centre O and radius equal to 1 × 4.6, i.e., 2.3 cm, draw arcs on either side 2 of PR, cutting XY at Q and S.

O P 4.6 cm

4 Join PQ, QR, RS and SP. PQRS is the required square.

Think and Tell Is there any other measure, other than the diagonal, that can help us construct a square?

Constructing a Rhombus A rhombus has lengths of all its sides equal, its opposite angles are equal and the diagonals bisect each other at right angles. Let us construct a rhombus whose diagonals are of lengths 5 cm and 7 cm.

It is known that the diagonals of a rhombus bisect each other at the right angle. This property can help us construct the rhombus. Let’s say, the rhombus is PQRS and the diagonals bisect at O.

Q 2.5 cm

PR = 7 cm, PO = OR = 3.5 cm (Diagonals bisect each other)

QS = 5 cm, QO = OS = 2.5 cm (Diagonals bisect each other)

cm O

3. P

S X

1 Draw a line segment PR = 7 cm. 2 Draw XY, its perpendicular bisector and let it intersect PR at O. 3 U sing O as centre and a radius of 2.5 cm, draw two arcs on the opposite sides of PR to intersect XY at Q and S.

2.5

Given:

Q O 2.5 cm R 7 cm

4 Join PQ, QR, RS and SP. PQRS is the required rhombus.

Constructing a Rectangle

We know that a rectangle has its opposite sides equal, all angles measure 90° and its diagonals are equal in length and bisect each other. Let us construct a rectangle PQRS, given that side PQ = 6.2 cm and the diagonal PR = 7.8 cm. We know that the opposite sides are equal and all the interior angles in the rectangle are right angles. This property can help us construct the rectangle. Let’s say, the rectangle is PQRS and PR is its diagonal. 68

7.8 P

6.2 cm

Given:

PR = 7.8 cm (Length of diagonal)

PQ = RS = 6.2 cm (Opposite sides of a rectangle are equal) ∠Q = 90° (All angles are right angles)

1 Draw a line segment PQ = 6.2 cm.

2 Draw XQ ⊥ PQ at point Q. 3 Using P as the centre and a radius of 7.8 cm, draw an arc cutting QX at R.

4 D raw two arcs: one with center R and a 6.2 cm radius, and another with center P and a radius equal to QR, ensuring they intersect at point S.

5 Join SP and SR. PQRS is the required rectangle.

We know that one pair of opposite sides are parallel in a trapezium. This property can help us construct the trapezium.

4.5 cm

cm 3.7

Let us construct a trapezium EFGH in which EF ॥ GH, HE = 3.7 cm, FE = 4.5 cm, GH = 6.7 cm and ∠H = 60° cm.

90°

6.2 cm

Constructing a Trapezium

We know that a trapezium has one pair of parallel sides and in an isosceles trapezium, the non-parallel sides are equal and the diagonals are equal.

60°

6.7 cm

Given: EF ॥ GH; HE = 3.7 cm, FE = 4.5 cm, GH = 6.7 cm.

1 Draw a line segment GH = 6.7 cm. E

4.5 cm

3.7

60°

2 Construct an angle of 60° at H. 3 Using H as centre and radius of 3.7 cm, draw an arc to cut XH at E. 4 Construct XY parallel to HG, passing through E.

6.7 cm

5 Using E as centre and radius of 4.5 cm, draw an arc to cut EY at F. 6 Join GF. EFGH is the required trapezium.

Constructing a Parallelogram

In a parallelogram, the opposite sides are equal and parallel, the opposite angles are equal and the diagonals bisect each other.

Let us construct a parallelogram one of whose sides is 6 cm and whose diagonals are 7 cm and 8 cm. D

3.5

It is known that diagonals of a parallelogram bisect each other. This property can help us construct the parallelogram. Let’s say, the parallelogram is ABCD and the diagonals bisect at O.

AB = 6 cm

AC = 8 cm, AO = OC = 4 cm (Diagonals bisect each other)

6 cm

Given:

3.5

BD = 7 cm, BO = OD = 3.5 cm (Diagonals bisect each other)

Chapter 5 • Construction of Quadrilaterals

1 Draw a line segment AB = 6 cm.

2 D raw two arcs: one with center A and a 3.5 cm radius, and another with center B and a 4 cm radius, intersecting at O.

3 Join OA and OB.

4 E xtend AO to meet C, making OC equal to AO and extend BO to meet D, making OD equal to BO. 5 Join AD, BC and CD. ABCD is the required parallelogram.

m 4c A

cm B

6 cm

Constructing a Kite

In a kite, two pairs of adjacent sides are equal and the diagonals intersect at right angles.

m 5c

It is known that the two pair of the adjacent sides have equal lengths in a kite. This property can help us construct the kite. Let’s say, WXYZ is a kite and WY is its diagonal.

m 5c

Let us construct a kite in which the measure of two adjacent sides are 5 cm and 3 cm with the length of the diagonal passing through the common vertex of these sides is 4 cm.

4 cm

Give: WY = 4 cm (Length of diagonal); WX = XY = 5 cm (adjacent sides are equal); WZ = YZ = 3 cm (Adjacent sides are equal)

Y 3

1 Draw a line segment WY = 4 cm.

4 D raw two more arcs: one with center at W (5 cm radius) and another with center at Y (5 cm radius), intersecting at X.

3 Join WZ and YZ.

2 D raw two arcs: one with center at W (3 cm radius) and another with center at Y (3 cm radius), intersecting at Z.

4 cm

5 Join WX and YX. WXYZ is the required kite.

Z Example 6

Construct a parallelogram PQRS in which PQ = 3 cm, QR = 4 cm and diagonal PR = 6 cm. It is known that the opposite sides of a parallelogram are equal. This property can help us construct a parallelogram. Let’s say, the parallelogram is PQRS and PR is its diagonal. Construct a parallelogram with its adjacent sides measuring 3 cm and 4 cm, respectively and the included angle being 60°.

6 P

3 cm

4 cm

Given: PQ = RS = 3 cm, RQ = SP = 4 cm, PR = 6 cm. Do It Together

Complete the given rough diagram of the parallelogram before starting the construction.

Error Alert! You may or may not able to construct a quadrilateral with any random set of measures of elements.

Rough Diagram

Construction

Do It Yourself 5B 1 Construct a square whose each side measures 5 cm. 2 Construct a rhombus ABCD whose side is 5 cm and an angle is 120°. 3 Construct a rectangle whose adjacent sides are 5.6 cm and 4.5 cm. 4 Construct a parallelogram with one side as 4.9 cm and diagonals as 5.4 cm and 8 cm. 5 Construct a parallelogram PQRS in which PQ = 4.6 cm, QR = 6.7 cm and ∠Q = 45°. 6 Construct a trapezium ABCD in which AB ॥ CD, AB = 5.8 cm, CD = 4.5 cm, ∠A = 67° and AD = 3.6 cm. What is the measure of the largest angle in the quadrilateral?

Word Problem 1

You are making a greeting card, and you want to create a square-shaped window in the

front cover. The diagonals of this window are 8 cm. You have a ruler and a compass. What steps will you follow to accurately draw a square on the card’s cover?

Points to Remember • A quadrilateral can be constructed if the measures of 5 out of its 10 elements are given. • Diagonals of parallelograms, rectangles, rhombuses and squares bisect each other. • A quadrilateral can be constructed when the measure of the 4 sides & 1 diagonal, 3 sides & 2 diagonals, 3 sides & 2 included angles, 3 angles & 2 included sides or 4 sides & 1 angle are given. • You need the knowledge of fewer elements to construct special quadrilateral as their properties help you determine other unknown elements. • The sum of the interior angles of a quadrilateral is 360°.

Chapter 5 • Construction of Quadrilaterals

Math Lab Quadrilateral Pattern Creation Materials Required: Drawing paper, rulers, compasses, pencils, protractors, coloured pencils/markers (optional) Instructions: 1 Start by reviewing the basic construction techniques for drawing quadrilaterals using a ruler and a compass. 2 Ask each student to brainstorm and plan a design or pattern that they want to create using quadrilaterals. They should consider the type of quadrilaterals they want to include (e.g., squares, rectangles, parallelograms, and so on) and the measurements for each side and angle. 3 Allow students to begin constructing their chosen patterns on their drawing paper. They should use their rulers, compasses and protractors. 4 Encourage students to get creative with their patterns. They can experiment with different colours and arrangements of the quadrilaterals to make their designs visually appealing. 5 Assess each student’s pattern on the basis of creativity, precision in construction and the ability to explain their design choices.

Chapter Checkup 1 Construct quadrilaterals with the given lengths of the 4 sides and 1 diagonal. a Quadrilateral UVWX: UV = 3.9 cm, VW = 5.1 cm, WX = 4.5 cm, XU = 6.8 cm, VX = 7.2 cm. b Quadrilateral STUV: ST = 4.5 cm, TU = 5.7 cm, UV = 6.2 cm, VS = 7.3 cm, SV = 8.9 cm. c Quadrilateral KLMN: KL = 6.1 cm, LM = 3.5 cm, MN = 5.8 cm, KN = 4.2 cm, LN = 7.4 cm.

2 Construct a quadrilateral PINE with the given lengths of 3 sides and 2 diagonals. Find the measure of the unknown side.

a PI = 6.9 cm, EN = 3.2 cm, IN = 5.5 cm, PN = 4.7 cm, IE = 7.6 cm. b PI = 7.4 cm, NE = 3.2 cm, PE= 5 cm, IE = 6.8 cm, PN = 4.7 cm c PI = 6.5 cm, IN = 4.2 cm, PE = 3.9 cm, PN = 7.1 cm, IE = 8.3 cm.

3 Construct the following quadrilaterals when the measure of the 3 angles and 2 included sides are given. Find the measure of both of its diagonals.

a Quadrilateral EFGH: EF = 6 cm, FG = 5 cm, ∠E = 60°, ∠F = 45°, and ∠G = 120° b Quadrilateral IJKL: IJ = 4.9 cm, IL = 7 cm, ∠I = 70°, ∠J = 90° and ∠L = 115° c Quadrilateral MNOP: MN = 6.5 cm, MP = 4.5 cm, ∠M = 100°, ∠N = 90° = and ∠P = 135°

4 Construct quadrilaterals WXYZ with the given measures. a WX = 4.5 cm, XY = 6.4 cm, YZ = 3.9 cm, ∠X = 150° and ∠Y = 60° b WX = 4.6 cm, XY = 5.5 cm, YZ = 6.6 cm, ∠X = 80° and ∠Y = 100° c WX = 3.2 cm, XY= 5.7 cm, YZ = 4.4 cm, ∠X = 135° and ∠Y = 60°

5 Construct the quadrilaterals given below where 4 sides and one of the angles are known. a Quadrilateral ABCD: AB = 4.6 cm, BC = 4.1, CD = 5.6 cm, AD = 5.2 cm and ∠B = 70° b Quadrilateral PQRS: PQ = 3.9 cm, QR = 3.3 cm, RS = 6 cm, PS = 5 cm and ∠Q = 80° c Quadrilateral WXYZ: WX = 7.6 cm, XY= 6.8 cm, YZ = 5.2 cm, ZW = 4.6 cm and ∠Y = 120°

6 Construct a square whose each diagonal measures 5.8 cm. 7 Construct a rhombus PQRS in which: a PQ = 4 cm and SQ = 6 cm

b QS = 6.6 cm and QR = 3.7 cm

8 Construct a rectangle ABCD when it is given that: a One side measures 4 cm and the diagonal has a length of 6.6 cm. b The diagonals measure 5.6 cm and the angle between them is 60°. c Its adjacent sides measure 3.7 cm and 2.9 cm, respectively.

9 Construct a parallelogram ABCD, given that BD = 4.6 cm, AC = 6 cm and the angle included between the diagonals is 60°.

10 Construct a trapezium EFGH in which EF ॥ GH, EF = 4.5 cm, BC = 3.7 cm, AD = 5 cm and ∠F = 40°. Can we construct more than one trapezium in this case?

11 Construct a parallelogram ABCD in which AC = 5.5 cm, AB = 3.6 cm and the altitude of AM from A to CD is 2.6 cm.

Word Problem 1

You are working on a craft project and need to create a rhombus-shaped outline with a

diagonal of 6 cm in length and the side of the length 4.2 cm. How can you use your ruler and compass to draw the rhombus accurately?

Chapter 5 • Construction of Quadrilaterals

3 Classification and 6 Tabulation of Data Let's Recall

We have learnt that data is collected and organised in tables, making it easier to understand. The given table shows the major crop-producing states in India. This collection of information is called data. From the above data set, we can conclude that: 3 of the states produce rice. 2 of the states produce wheat. 2 of the states produce pulses. 1 of the states produces maize. So, we can say that rice is the most produced crop in India.

State

Crop Produced

Haryana

Wheat

Odisha

Rice

West Bengal

Rice

Uttar Pradesh

Wheat

Rajasthan

Pulses

Tamil Nadu

Rice

Madhya Pradesh

Pulses

Karnataka

Maize

Letʼs Warm-up

The given table shows the temperatures in a month every day for 30 days. Calender

MON 1

Calender Calender Calender Calender Calender Calender TUE WED THU FRI SAT SUN Calender Calender Calender Calender Calender Calender Calender Calender Calender 2 Calender 3 4 5 6 7 Calender Calender Calender Calender Calender

Calender 8

Complete the tally chart.

Calender Calender

Calender Calender Calender Calender

Temperature (in ˚C) 20

Tally

Calender

Total

25 30 35

I scored _________ out of 4.

Frequency Distribution of Grouped and Ungrouped Data Real Life Connect

Ahan loves reading books. Every time he finishes reading a book, he writes down its title, author’s name, and how much he likes it on a scale of 10. He writes them in his diary as: Book Title

Author

Rating Out of 10

Number of Pages

Summer’s Child

Diane Chamberlain

416

Sanctuary

Caryn Lix

320

The Diabolic

S. J. Kincaid

416

Last Star Burning

Caitlin Sangster

400

Rabbit and Robot

Andrew Smith

448

We now know that the details that Ahan collects in his reading log is the data. Data is a collection of numerical figures that represents a specific kind of detail called an observation. The above data is recorded in a table in an unorganised form. When numerical data is collected in its original form and in an unorganised form, the data is called ungrouped data or raw data. We use data to remember information about all sorts of things in our everyday lives, like keeping track of grades in school, the number of goals scored in sports, or even the weather forecast throughout the day. This study of numerical data is called statistics. The stages of statistical study are: 1 Collection of data

Think and Tell

2 Organisation and presentation of data

Where do you need to organise data in your day-to-day life?

3 Analysis and interpretation of data Types of Data Qualitative Data

Quantitative Data

It is descriptive, i.e., it describes the quality of data.

It refers to the information that can be counted or measured, i.e., it can be assigned a numerical value.

For example, the qualitative data in Ahan’s reading log refers to the rating he gave to each book out of 10.

For example, the quantitative data in Ahan’s reading log includes information like the number of pages in each book.

When we organise raw data by arranging them in rows or columns, it is called an array. The data can be arranged either in ascending order or descending order. An array is like a bookshelf in Ahan’s room where he keeps his books neatly arranged, with each book representing a piece of data. So, in Ahan’s reading log, an array would be a list that holds all the data about the books.

Chapter 6 • Classification and Tabulation of Data

Organising Data Organising data is like arranging your books neatly on a bookshelf or putting them in specific categories. In the reading log table, Ahan has organised data about five books he has read.

Did You Know? Ungrouped data is also called discrete data.

This format makes it easy for Ahan to track the books read and find specific information about each book. By organising data, you can quickly access the information you need and then discover trends or patterns.

Frequency Distribution of Ungrouped Data We saw how Ahan tracks book genres like mystery, horror, and science fiction in his reading log. This helps to know how many times each genre appears, which genres Ahan reads most frequently, and which are less common. The number of times a number or an observation occurs in a given set of data is called frequency. Let us create a frequency distribution and record the number of pages in each book he has read. Book Title

Summer’s Child

Sanctuary

The Diabolic

Last Star Burning

Rabbit and Robot

Number of Pages

416

320

416

400

448

Step 1: List all the unique values (the different numbers of pages) from Ahan’s reading log as 416, 320, 416, 400, and 448. Step 2: Count how many times each unique value appears in the data. This gives us the frequency. 416: 2 books; 320: 1 book; 400: 1 book; 448: 1 book

Step 3: Create a table that displays the unique values and their corresponding frequencies. Number of Pages

Frequency

Tally Marks

416

320

400

448

This table is a frequency distribution of the ungrouped data which shows how many times each specific number of pages appeared in Ahan’s reading log. Ahan read two books with 416 pages, one book with 320 pages, another with 400 pages, and a final one with 448 pages. If Ahan records the number of pages in each book he has read, the range is the difference in the number of pages between the thinnest and thickest books. The range refers to the difference between the smallest and largest values for a piece of information. For Example, Sarah recorded the marks her classmates scored out of 10 on a recent test. 5, 5, 7, 8, 9, 10, 8, 9, 7, 10, 9, 5, 8, 7, 8, 9, 10, 10, 5, 8

Array the data and form a frequency table. On arranging the data in ascending order, we get: 5, 5, 5, 5, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10 The frequency distribution table can be drawn as: Marks

Tally Marks

Number of students (Frequency)

||||

|||

||||

Total

Studies suggest that on an average, students should spend about 5–10 hours per week on extra-curricular activities. Given below are the number of hours 30 students spend on extra-curricular activities per week.

Example 1

5, 6, 7, 3, 5, 4, 7, 6, 5, 3, 7, 6, 4, 5, 6, 5, 6, 7, 3, 5, 4, 7, 6, 5, 3, 7, 6, 4, 5, 6

Array the data from a frequency table and find out the range of the given data.

Step 1: List the unique values in ascending order.

We get: 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7

Step 2: Count the Frequency and create the Frequency Distribution Table. Hours spent on extra-curricular activities

Tally marks

Number of students (Frequency)

||||

|||||||

|||||

Total

6 students spent the maximum number of hours on extra-curricular activities. 4 students spent the minimum number of hours on extra-curricular activities. Therefore, the range of the given data = Maximum value Minimum value =3-2=1

Do It Together

Suppose we have data representing the number of siblings that 30 individuals have. The data set is as follows: 2, 3, 4, 2, 1, 3, 0, 2, 4, 3, 2, 1, 1, 0, 0, 0, 1, 3, 4, 2, 2, 2, 1, 1, 3, 1, 1, 0, 2, 0

Array the data and form a frequency table. Also, find out the range of the given data. Number of Siblings

Tally Marks

Frequency

|||||

|||| |||

_________

Total Chapter 6 • Classification and Tabulation of Data

_________

Number of individuals that have a maximum number of siblings = _______________ Number of individuals that have a minimum number of siblings = _______________ Therefore, the range of the given data = _______________

Do It Yourself 6A 1 In a class, there are 50 students. Each student was asked what their favourite sport was. 40% of the students said cricket, 25% said badminton, 20% said tennis, 10% said football, and the rest did not have any favourite sport.

Prepare a frequency distribution table for the number of students who like a sport. Which sport is liked by the minimum number of students?

2 This table is the frequency distribution that shows the monthly income levels of middle-class families in India. Monthly Income (in thousands) (in ₹)

Frequency

a How many individuals have the highest annual income? b How many individuals have the lowest annual income? c What is the range?

3 The given data represents the ages of 20 people in a neighbourhood: 25, 30, 35, 40, 25, 28, 35, 42, 30, 32, 28, 22, 25, 28, 40, 45, 35, 30, 32, 28 Prepare a frequency distribution table for the above data.

4 The given data represents the scores (out of 100) of 30 students on a maths test. The data set is as follows: 85, 92, 78, 90, 88, 76, 95, 82, 79, 88, 94, 87, 89, 91, 75, 83, 80, 92, 84, 86, 77, 81, 93, 79, 88, 90, 86, 85, 94, 87 Array the data and form a frequency table. Also, find out the range of the given data.

5 Suppose you have data representing the number of holidays taken in a year by 30 individuals. The holiday data is as follows:

2, 3, 2, 5, 3, 4, 1, 3, 5, 2, 3, 4, 2, 1, 3, 4, 2, 5, 1, 2, 3, 4, 1, 4, 5, 1, 2, 4, 1, 5 Array the data and form a frequency table. Also, find out the range of the given data.

6 The value of π up to 39 decimal places is 3.141592653589793238462643383279502884197 Make a frequency distribution table of the digits 0 to 9 before and after the decimal point.

Grouping Data COVID-19, initially detected in India in early 2020, has had a profound impact on the nation’s healthcare system, economy, and daily life, prompting extensive vaccination campaigns and public health measures. The given table shows the number of COVID-19 cases in various countries around the world.

Country

Number of COVID-19 Cases (in crores)

France

India

Germany

Italy

United Kingdom

United States

Spain

Brazil

Japan

Turkey

While creating a frequency distribution for this data, we tracked the number of COVID cases for each country separately. This approach is useful when we need to analyse large data. However, it is sometimes beneficial to group data to get a broader overview. One common way to group data is by using ranges or intervals. For instance, we can group the number of COVID cases. The number of cases was in ranges such as 0–2 crores and 3–5 crores cases. Grouping data can make it easier when we need to deal with large data. When individual observations are arranged in groups such that a frequency distribution table of these groups helps provide a convenient way of presenting or analysing data is known as grouped data.

Drawing a Frequency Distribution Table of Grouped Data Let us group the number of COVID cases into the ranges 0–2, 3–5, 6–8, and 9–11. These ranges are called class intervals. When the whole range of variable values is classified in some groups in the form of intervals, then each such interval is known as a class interval. The range of a class interval is called the class limit. The smaller value of a class interval is called the lower class limit, and the greater value is called the upper class limit. For example, in the class interval 3–5, 3 is the lower class interval and 5 is the upper class interval. The difference between the upper limit and the lower limit of a class interval is called the class size. For the class interval 3–5, the class size is 2. The class mark is the midpoint of each class interval. For the class interval 3-5, the class mark is 3 + 5 = 4. 2 Chapter 6 • Classification and Tabulation of Data

Did You Know? Sir Ronald Fisher introduced the concept of data handling or statistics. Indian mathematicians P.C. Mahalanobis and C. R. Rao have also played a major role in the field of statistics.

Remember! class mark =

upper limit + lower limit 2

Remember!

Error Alert!

When we take the sum of all the frequencies, it should be equal to the total number of observations.

The class size of all the class intervals must not differ.

Types of Frequency Distribution Exclusive Form (Continuous Form) Exclusive class intervals can be defined as 10–15, 15–20, 20–25, and so on. In this interval, the upper limit of one class is the lower limit of the next class.

In this form, the lower and upper limits are known as the true lower limit and the true upper limit of the class interval.

Inclusive Form (Discontinuous Form) Inclusive class intervals can be defined as 9–14, 15–20, 21–26, and so on.

In this interval, the upper limit as well as the lower limit are included in the class.

Listed below are the steps to identify the rules of choosing class intervals for a grouped frequency distribution table. 1 Identify the highest and lowest data values and find the difference between them. 2 Decide the number of class intervals needed (between 5 to 15 classes). 3 Draw the frequency distribution table to show the data. Based on the above information, we can form a frequency distribution table as follows: Number of COVID cases in each Country (in crores) (Range)

Tally Marks

Frequency

0–2

|||

3–5

||||

6–8

9–11

Reading a Frequency Distribution Table of Grouped Data In the above table, we grouped the data by ranges and counted the number of COVID cases falling into each range. This grouped frequency distribution provides a concise summary of the number of cases all around the world. Let us now read what information the table provides.

•

There were 0–2 crore COVID cases in 3 countries.

•

There were 3–5 crore COVID cases in 5 countries.

•

There were 6–8 crore COVID cases in 1 country.

•

There were 9–11 crore COVID cases in 1 country.

The marks of 30 students in a science test are given below.

Example 2

Class Intervals

Tally Marks

Frequency

0–5

5–10

|||

10–15

|||| |||

15–20

|||| |

20–25

||||

25–30

|||| ||

Total

Answer the following questions. 1 What is the lower limit of the first class interval?

2 What is the upper limit of the last class interval? 30 3 What is the size of each class?

4 Which class interval has the highest frequency? 5 Which class interval has the lowest frequency?

(10 - 15) (0 - 5)

6 What is the class mark of the second class interval? Example 3

5 + 10 15 = = 7.5 2 2

Read the frequency distribution table showing the ages of 50 people in a neighbourhood. Answer the questions that follow. Class Intervals

Tally Marks

Frequency

0–10

|||

10–20

|||| |||| |

20–30

|||| |||| |||

30–40

|||| |||| ||||

40–50

|||| |||

Total

1 What is the upper limit of the first class interval? 10 2 What is the upper limit of the last class interval? 50 3 What is the size of each class? 10 4 Which class interval has the highest frequency?

(30 - 40)

5 Which class interval has the lowest frequency? (0 - 10) 30 6 What is the class mark of the second class interval? 10 + 20 = = 15 2 2

Chapter 6 • Classification and Tabulation of Data

Do It Together

The data below shows the annual income (in thousands) (in ₹) of 40 families in a town. 71

Prepare a frequency distribution table for the data using an appropriate scale and then answer the following questions: 1 What is the lower limit of the first class interval?

_________

2 What is the upper limit of the last class interval?

_________

3 What is the size of each class?

_________

4 Which class interval has the highest frequency?

_________

5 Which class interval has the lowest frequency?

_________

6 What is the class mark of the fifth class interval?

_________

Do It Yourself 6B 1 The following is the distribution of weights (in kg) of 46 persons: i

What is the lower limit of class 50–60?

Find the class marks of the classes 40–50 and 50–60.

iii

What is the class size?

2 Look at the given observations. These figures show the percentage of crops that were harvested in various seasons.

53, 61, 48, 60, 78, 68, 55, 100, 67, 90, 75, 88, 77, 37, 84, 58, 60, 48,

Weight (in kg)

Persons

a 30–40

b 40–50

c 50–60

d 60–70

e 70–80

62, 56, 44, 58, 52, 64, 98, 59, 70, 39, 50, 60

a Arrange these observations in ascending order, say 30 to 39 as the first group, 40 to 49 as the second group, and so on.

Now, answer the following: b What is the highest observation?

c What is the lowest observation?

d What is the range?

e How many observations are 75 or more?

How many observations are less than 50?

3 The observations given below show the ages of people who watch a particular T.V. channel. Prepare a grouped frequency table for the data:

20, 39, 42, 5, 12, 19, 47, 27, 7, 13, 40, 38, 24, 34, 15, 40, 10, 9, 3, 29, 17, 34, 23, 18, 19

4 Look at the given frequency distribution table. Number of Siblings

0–1

2–3

4–5

6–7

8–9

Frequency

Answer the following questions: a How many individuals in the data set have 2 or 3 siblings? b What is the total number of individuals with 4 or 5 siblings? c Calculate the percentage of individuals with 0–1 sibling. d Find the range of the number of siblings.

5 The given table shows the marks obtained (out of 20) by students in a class test. Read the table and answer the following questions. Marks

0–5

5–10

10–15

15–20

Frequency

a How many students scored marks more than or equal to 20? b How many students scored more than 15 marks? c What is the class mark of the class interval with frequency 6? d What percentage of students scored less than 15 marks? e How many students write the test?

Word Problems 1 The marks obtained by 27 students of a class in an examination are given below:

23, 18, 13, 10, 21, 7, 1, 13, 21, 13, 15, 24, 16, 3, 23, 5, 6, 8, 7, 9, 12, 20, 10, 2, 23, 24, 0

Draw a grouped frequency distribution table.

2 The weight of 25 students in a class was recorded. Form an ungrouped frequency distribution table for the data given below and answer the following questions.

25, 24, 20, 25, 16, 15, 18, 20, 25, 16, 20, 16, 15, 18, 25, 16, 24, 18, 25, 15, 27, 20, 20, 27, 25. i

Find the range of the weights.

How many of the students have the maximum weight in the class?

iii

What is the weight of the maximum students?

How many students have the least weight in the class?

3 The heights of 25 students, measured in centimetres, were recorded as follows: 155, 149, 160, 162, 154, 149, 157, 158, 163, 171, 170,152, 155, 163, 172, 162, 162, 154, 159,

161, 171, 173, 149, 157, 155

Represent the above data by a grouped frequency distribution table, using tally marks.

4 The electricity bill (in ₹) for each of the 24 houses in a village is given below. Construct a frequency table.

215, 203, 120, 350, 800, 600, 350, 400, 120, 340, 150, 562, 452, 125, 658, 235, 645, 450, 207, 489, 263, 500, 153, 450

Chapter 6 • Classification and Tabulation of Data

Points to Remember • Collection of numerical figures that represent a similar kind of information is called data. • A table is called a frequency distribution table when it shows how many times each specific number or figure appears in a data set. • In grouped data, individual observations are arranged into groups such that a frequency distribution table of these groups helps provide a convenient way of presenting or analysing the data. • We follow these basic steps when drawing a frequency distribution table for a data set that contains multiple observations: 1 Find the lowest and highest values of the variables. 2 Decide the size of the class intervals. 3 Write the frequency of each interval.

Math Lab Setting: In pairs Materials: A dice, a pen, and a sheet of paper Method:

Divide the students into pairs.

Each pair rolls the dice 15 times.

Record the observation after each roll.

Create a frequency distribution table to record the observations

The first pair to complete the frequency table of the 15 observations wins.

Chapter Checkup 1 Construct a frequency table for each of the following data set. a 4, 3, 6, 5, 2, 4, 3, 3, 6, 4, 2, 3, 2, 2, 3, 3, 4, 5, 6, 4, 2, 3, 4 b 6, 7, 5, 4, 5, 6, 6, 8, 7, 9, 6, 5, 6, 7, 7, 8, 9, 4, 6, 7, 6, 5

2 Represent the following data in the form of a frequency distribution. 1, 2, 6, 5, 1, 5, 1, 3, 2, 6, 2, 3, 4, 2, 0, 0, 4, 4, 3, 2, 2, 0, 0, 1, 2, 2, 4, 3, 2, 1, 0, 5, 1, 2, 4, 3, 4, 1, 6, 2, 2.

3 A die was thrown 25 times, and the following scores were obtained: 6, 4, 2, 5, 3, 3, 1, 2, 6, 1, 3, 1, 2, 3, 4, 5, 6, 6, 2, 4, 4, 4, 5, 6, 1 Prepare a frequency table of the scores.

4 A survey was conducted among students, and the observation below shows that the students start preparing for

the entrance examinations when they are 12–18 years’ old. Prepare a frequency table based on the following ages: 13, 14, 13, 12, 14, 13, 14, 15, 13, 14, 13, 14, 16, 12, 14, 13, 14, 15, 16, 13, 14, 13, 12, 17, 13, 12, 13, 13, 13, 14

5 The given table shows the number of blood donors of each blood group in a hospital’s blood donation campaign. Blood group

Frequency

a Which blood group has the most donors?

b How many donors have the AB blood group?

6 Complete the table given below: Class limit

Class interval

Lower limit

Upper limit

Class size

Class marks

0–10 10–20 20–30 30–40 40–50

Word Problems 1

The population of some Indian states are given in the table below. State Population (in crores)

Meghalaya Maharashtra Bihar Andhra West Nagaland Uttarakhand Pradesh Bengal 3

a How many class intervals are there in the table? b Which is the most populated state? c Which is the least populated state? 2

survey was taken across 20 houses, and the residents were asked about the number of cars A registered in their households. The results were recorded as follows: 4, 1, 4, 0, 2, 1, 1, 2, 1, 0, 4, 2, 3, 2, 0, 2, 1, 0, 3, 2 Present this data in a frequency distribution table. Also, find the maximum number of cars registered by a household.

Dr. Radha recorded the pulse rate (per minute) of 29 persons: 61, 75, 71, 72, 70, 65, 77, 72, 67, 80, 77, 62, 71, 74, 79, 67, 80, 77, 62, 71, 74, 61, 70, 80, 72, 59, 78, 71, 72 She wants to construct a frequency table in the inclusive form, taking the class interval 61–65 of equal width. Now, if she needs to convert this data again into the exclusive form in a separate table, how can she do that?

Chapter 6 • Classification and Tabulation of Data