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