Basic Mathematics for College Students

Pythagorean theorem

If and are the lengths of the legs of a right triangle

and is the length of the hypotenuse, then

a^2 b^2 c^2

c

a b

DEFINITIONS AND CONCEPTS EXAMPLES

c

a

b

Leg

Leg

Hypotenuse

Find the length of the hypotenuse

of the right triangle shown here.

We will let and , and

substitute into the Pythagorean

equation to find.

This is the Pythagorean equation.

Substitute 6 for aand 8 for b.

Evaluate the exponential expressions.

Do the addition.

Reverse the sides of the equation so that c^2 is

on the left.

To find , we must find a number that, when squared, is 100. There

are two such numbers, one positive and one negative; they are the

square roots of 100. Since represents the length of a side of a

triangle, cannot be negative. For this reason, we need only find

the positive square root of 100 to get.

The symbol is used to indicate the postive

square root of a number.

Because 10^2 100.

The length of the hypotenuse of the triangle is 10 in.

c 10

c 1100 1

c

c

c

c

c^2  100

100 c^2

36  64 c^2

62  82 c^2

a^2 b^2 c^2

c

a 6 b 8

8 in.

6 in.

820 Chapter 9 An Introduction to Geometry

When we use the Pythagorean theorem to find the

length of a side of a right triangle, the solution is

sometimes the square root of a number that is not a

perfect square. In that case, we can use a calculator to

approximatethe square root.

The lengths of two sides of a right triangle are

shown here. Find the missing side length.

We may substitute 9 for either or , but 11

must be substituted for the length of the

hypotenuse. If we substitute 9 for , we can find the unknown side

length as follows.

This is the Pythagorean equation.

Substitute 9 for aand 11 for c.

Evaluate each exponential expression.

To isolate b^2 on the left side,

subtract 81 from both sides.

We must find a number that, when squared, is 40. Since

represents the length of a side of a triangle, we consider only the

positive square root.

This is the exact length.

The missing side length is exactly feet long. Since 40 is not a

perfect square, we use a calculator to approximate. To the

nearest hundredth, the missing side length is 6.32 ft.

140

140

b 140

b

b^2  40

81 b^2  81  121  81

81 b^2  121

92 b^2  112

a^2 b^2 c^2

b

a

c

a b

9 ft 11 ft

SECTION 9.4 The Pythagorean Theorem

a^2 b^2 c^2 is called the Pythagorean equation.