The Pythagorean Theorem
A 90Â° angle is a right angle, so a triangle that contains a 90Â° angle is
called a right triangle. The side opposite the right angle is the
hypotenuse of the triangle.
Let a and b represent the lengths of the sides that form the right angle. Let
c represent the length of the hypotenuse.
The Pythagorean Theorem states the relationship between the lengths of
the sides of a right triangle.
Formula â€”
Pythagorean Theorem
If a and b are the lengths of the sides that form the right angle in a
right triangle and c is the length of the hypoteneuse, then:
c^{2} = a^{2} + b^{2}
If we know the lengths of two sides of a right triangle, we can use the
Pythagorean Theorem to find the length of the third side.
Example 1
Given a right triangle with sides of length 6 inches and 8 inches, find the
length of the hypotenuse.
Solution
In the Pythagorean Theorem, substitute
6 inches for a and 8 inches for b.

c^{2} c^{2} 
= a^{2}
+ b^{2}
= (6 inches)^{2} + (8 inches)^{2}

Calculate the squares.
Add.
The value of c^{2} is 100.


= 36 inches^{2}
+ 64 inches^{2}
= 100 inches^{2} 
To find the value of c we ask,
â€œWhat positive number squared is 100?â€
That is, â€œWhat is the square root of 100?â€
The square root of 100 is 10. 
c
c 
= 10 inches 
Therefore, the length of the hypotenuse, c, is 10 inches.
Example 2
Given a right triangle where one leg has length 3 units and the hypotenuse
has length 5 units, find the length of the third side.
Solution
In the Pythagorean Theorem, substitute 3
for a and 5 for
c.

c^{2}
5^{2} 
= a^{2}
+ b^{2}
= (3)^{2} +
b^{2} 
Calculate the squares.
Subtract 9 from both sides. 
25
16 
=
9
+ b^{2} =
b^{2} 
We found that b^{2} is 16.
The square root of 16 is 4. 
4 
= b 
Therefore, the length of the third side, b, is 4 units.
