Pythagoras

Pythagoras Pythagoras was a famous Greek Mathematician who discovered a connection between the 3 sides of any right-angled triangle

570BC - 495BC

3

2

4

5

Investigate the relationship 1

Pythagoras The Theorem of Pythagoras Ar

ea

1

Area 3 = Area 1 + Area 2 Area 2

a

b

therefore

2

2

2

c = a + b

c

Area 3

We can know calculate one side of a right-angled triangle if we know the lengths of the other two

Pythagoras Naming the sides of a triangle

ctenuse

o p y

H

b

a The longest side of a right-angled triangle is called the hypotenuse. Its always opposite the right angle

Pythagoras Lets look at how to use Pythagoras' Rule The two smaller sides of the right angled triangle are 8cm and 6cm

c

b

6cm

a

8cm

To calculate the length of the hypotenuse

c2 2 c c2 c

= = = =

a2 + b2 82 + 62 100 √100 = 10cm

Pythagoras

Hint

We can also use Pythagoras to calculate one of the smaller sides

12cm

7cm

x To calculate the length of a smaller side

c2 x2 x2 x

= = = =

a2 + b2 2 2 12 - 7 Subtract 95 √95 = 9.75cm

(2 d.p.)

Pythagoras Problem solving using Pythagoras Theorem Whenever a problem involves trying to find the missing side of a right-angled triangle consider Pythagoras' Theorem

2

2

2

c = a + b

p

ram

1.6m

r2 = 8.22 + 1.62 2

r = 69.8 r = √69.8 = 8.4m

8.2m Shown above is a design for a stunt bike ramp Calculate the length of the ramp

Pythagoras Problem solving using Pythagoras Theorem 50cm

82cm

65cm

Can I prove this door has right-angles at the corners?

x2 = 502 + 652

x2 = 6725 x = √6725 = 82cm From Pythagoras the door must be right-angled

The CONVERSE of Pythagoras We can use Pythagoras Theorem in reverse Prove that triangle ABC is right­angled

AB = 6.8 BC = 5.1

AC = 8.5

B

AB2 = 46.24 BC2 = 26.01 2

AC = 72.25

5.1cm

6.8cm

C A

8.5cm

AB2 + BC2 = 72.25 = AC2 By the converse of pythagoras' theorem the triange is right­angled at B

Pythagoras Problem solving using Pythagoras Theorem

Calculate the length of the roof 2.5m

roof Working

c2 = a2 + b2 7.6m

r2 = 4.32 + 2.52 5.1m

r2 = 24.74 r = √24.74 = 5m

4.3m

Pythagoras Harder Problem solving using Pythagoras Theorem Gold chains are displayed diagonally on a square board of side 20inches. Calulate the length of the longest chain.

c2 = a2 + b2 c2 = 202 + 202 c2 = 800

c = √800 = 28.3inches 20 inches

Pythagoras Harder Problem solving using Pythagoras Theorem P

PQRS is a rhombus

Calculate the length of side PQ

8cm

Q

(Round answer to 1 d.p.)

S

3.5cm

16cm

c2 = a2 + b2

PQ2 = 3.52 + 82 R 7cm

PQ2 = 76.25 PQ = √76.25 = 8.7cm

Pythagoras Harder Problem solving using Pythagoras Theorem Find the distance between point A and point B y

B(8,7)

x2 = 62 ­ 42 4

7­3

A(2,3) 0

8­2

6 X

x2 = 20 x = √20

x = 4.47

Pythagoras (in 3D) Using pythagoras can we find the distance form F to D?

A

H

D

7cm

B

C

7cm

F

7cm

G

7cm

F

7cm

G

F

9.9cm

FG2 = 98

FG = √98 = 9.9cm

D

FD2 = 72 + 9.92

7cm

FD2 = 147.01

H

E

FG2 = 72 + 72

H

FD = √147.01

FD = 12.12cm