Pythagoras' Theorem

We call it a right triangle
because it contains a
right angle.

The measure of a right angle
is 90o
90o

The little square
90o
in the
angle tells you it is a
right angle.

About 2,500 years ago, a
Greek mathematician named
Pythagorus discovered a
special relationship between
the sides of right triangles.

Pythagorus realized that if
you have a right triangle,
3
4
5

and you square the lengths
of the two sides that make
up the right angle,
2
4
2
3
3
4
5

and add them together,
3
4
5
2
4
2
3 2
2
4
3 

2
2
4
3 
you get the same number
you would get by squaring
the other side.
2
2
2
5
4
3 

3
4
5

Is that correct?
2
2
2
5
4
3 

?
25
16
9 

?

The side opposite from the
right angle
a
b
is called the

And the length of the
hypotenuse
is usually labeled c.
a
b
c

The relationship Pythagoras
discovered is now called
The Pythagoras’ Theorem:
a
b
c

The Pythagoras’ Theorem
says, given the right triangle
with legs a and b and
hypotenuse c,
a
b
c

then
a
b
c
.
2
2
2
c
b
a 


You can use The Pythagoras’
Theorem to solve many kinds
of problems.
Suppose you drive directly
west for 8 miles,
8

Then turn south and drive for
6 miles.
8
6

How far are you from where
you started?
8
6
?

82
Using The Pythagoras’
Theorem,
8
6
c
62
+ = c2

Why?
Can you see that we have a
right triangle?
8
6
c
82
62
+ = c2

Which side is the hypotenuse?
8
6
c
82
62
+ = c2

2 2
8 6
 
Then all we need to do is
calculate:
64 36 100
 
100  2
c

And you end up 10 miles from
where you started.
8
6
10
So, since c2
is 100, c is 10.

Find the length of a diagonal
of the rectangle:
15"
8"
?

of the rectangle:
15"
8"
?
b = 8
a = 15
c

2
2
2
c
b
a 

2
2
2
8
15 c


2
64
225 c


289
2

c
2
289
17
c
c


b = 8
a = 15
c

of the rectangle:
15"
8"
17

Exercise 10A
Q.1: Find the value of the unknown in each of the following right-
angled triangles.
(a)
20 cm
21 cm
a cm

(a) Solution:
Using Pythagoras’theorem,
a2 = 202 + 212
a2 = 400 + 441
a2 = 841
𝑎2 = 841
a = 29 cm
20 cm
21 cm
a cm
Hypotenuse

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