Basic Mathematics for College Students

9.4 The Pythagorean Theorem 747

SECTION 9.4

The Pythagorean Theorem

A theoremis a mathematical statement that can be proven. In this section, we will
discuss one of the most widely used theorems of geometry—the Pythagorean
theorem. It is named after Pythagoras, a Greek mathematician who lived about 2,500
years ago. He is thought to have been the first to develop a proof of it. The
Pythagorean theorem expresses the relationship between the lengths of the sides of
any right triangle.

Objectives

1 Use the Pythagorean theorem

to find the exact length of a

side of a right triangle.

2 Use the Pythagorean theorem

to approximate the length of a

side of a right triangle.

3 Use the converse of the

Pythagorean theorem.

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Pythagoras

1 Use the Pythagorean theorem to find the

exact length of a side of a right triangle.

Recall that a right triangle is a triangle that has a
right angle (an angle with measure 90°). In a right
triangle, the longest side is called the hypotenuse.It
is the side opposite the right angle. The other two
sides are called legs.It is common practice to let the
variable represent the length of the hypotenuse
and the variables and represent the lengths of the
legs, as shown on the right.
If we know the lengths of any two sides of a right triangle, we can find the length
of the third side using the Pythagorean theorem.

a b

c

Pythagorean Theorem

If and are the lengths of two legs of a right triangle and is the length of

the hypotenuse, then

a^2 b^2 c^2

a b c

In words, the Pythagorean theorem is expressed as follows:

In a right triangle, the sum of the squares of the lengths of the two legs is equal

to the square of the length of the hypotenuse.

Caution! When using the Pythagorean equation , we can let

represent the length of either leg of the right triangle. We then let represent

the length of the other leg. The variable must always represent the length of

the hypotenuse.

c

b

a^2 b^2 c^2 a

a c

b

Leg

Leg

Hypotenuse

EXAMPLE (^1) Find the length of the hypotenuse of
the right triangle shown here.
StrategyWe will use the Pythagorean theorem to find the
length of the hypotenuse.
WHYIf we know the lengths of any two sides of a right triangle, we can find the
length of the third side using the Pythagorean theorem.
Self Check 1
Find the length of the hypotenuse
of the right triangle shown below.
5 ft
12 ft
3 in.
4 in.