10-4 The Pythagorean Theorem

Algebra 1 Glencoe McGraw-Hill Jo Ann Evans

If you do not have a calculator, please get one from the back wall! The Chapter 10 test is a NON calculator test!

This is Pythagoras, a Greek mathematician who lived from about 585-500

B.C.

Although the Pythagorean theorem is named after him, there are indications that this theorem was in use in northern Africa before Pythagoras wrote of it.

Any triangle that has a right angle (90°) is called a right triangle. The two sides that meet to form the right angle are called the legs. The side across from the right angle is called the hypotenuse.

(hypotenuse)

(leg)

(leg)a

b

c

Hypotenuse, not Hippotenuse!

It doesn’t matter which leg is “a” and which leg is “b”, but the hypotenuse “c” is always the side directly across from the right

angle.

a2 + b2 = c2

For any right triangle, the sum of the squares of the legs of the

triangle is equal to the square of the hypotenuse.

The Pythagorean Theorem is a statement that describes the relationship among

the three sides of a right triangle.

(hypotenuse)

(leg)

(leg)a

b

c (leg)2 + (leg)2 = (hypotenuse)2

If necessary, round answers to the nearest hundredth. The Chapter 10 test is a NON calculator test!

In a right triangle, the lengths of the legs are 4 cm and 6 cm. Find the length of the hypotenuse.Draw a diagram.

4

6

c(leg)2 + (leg)2 = (hypotenuse)2

a2 + b2 = c2a =

b =

It doesn’t matter which leg is “a” and which leg is “b”, but the hypotenuse “c” is

always the side directly across from the right

angle.

222 c64 2c3616 2c52

2c52

Length will be a

positive number, so find

only the positive square root.

c132

132

The length of the hypotenuse is cm or about 7.21 cm.

132

c52

In a right triangle, the length of a leg is 13 cm and the length of the hypotenuse is 17 cm. Find the length of the other leg. Draw a diagram.

13

c =

(leg)2 + (leg)2 = (hypotenuse)2

a2 + b2 = c2

a =

bIt doesn’t matter which

leg is “a” and which leg is “b”, but the hypotenuse “c” is

always the side directly across from the right

angle.

222 17b13 289b169 2 169169

120b2 120b2

17

120b 302b

Length is cm or about 10.95 cm.

302

Example 1 In a right triangle, the lengths of the legs are 6 cm and 8 cm. Find the length of the hypotenuse.Draw a diagram.

6

8

c(leg)2 + (leg)2 =

(hypotenuse)2

a2 + b2 = c2

a =

b =It doesn’t matter which

leg is “a” and which leg is “b”, but the hypotenuse “c” is

always the side directly across from the right

angle.

222 c86 2c6436 2c100

2c100c10

= 10

The length of the hypotenuse is 10 cm.

Example 2 In a right triangle, the length of a leg is 12 cm and the length of the hypotenuse is 15 cm. Find the length of the other leg.

Draw a diagram.

12

c =

(leg)2 + (leg)2 = (hypotenuse)2

a2 + b2 = c2

a =

b222 15b12

225b144 2 144144

81b2

9b

15

9

The length of the other leg is 9 cm.

81b2

Example 3 Find the diagonal length of a TV screen that is 10 in wide by 20 inches long.

20

10

a2 + b2 = c2

222 c2010 2c400100 2c500

2c500

c510 The diagonal length of the TV screen is inches or about 22.36 inches.

c

510

Example 4 Find the unknown length.

Be careful! Remember to square the whole side.

3x8”

a2 + b2 = c2 10”

222 10)x3(8

100x964 2

2x

36x9 2

4x2 If x = 2, then the unknown length is 3(2).

The leg is 6” long.

Example 5 What is the longest line you can draw on a poster that is 15 inches by 25 inches?

15

25

a2 + b2 = c2

c15.29

222 c2515

2c625225 2c850

The longest line possible would be about 29.15 inches long.

Example 6 Solve for x to find the missing lengths of

the right triangle. x + 2

x

x +1

a2 + b2 = c2

1xor3x

222 )2x()1x(x

4x4x1x2xx 222 4x4x1x2x2 22

03x2x2 0)1x)(3x( The lengths of the

triangle are 3, 4, and 5.

Example 7 A right triangle has one leg that is 2 inches longer than the other leg. The hypotenuse is 10 inches. Find the unknown lengths.

x10

a2 + b2 = c2

x + 2

222 10)2x(x

1004x4xx 22

1004x4x2 2 100100 096x4x2 2

0)6x)(8x(2 6xor8x

a

b

cSubstitute.

Simplify.

Write in standard form.

Factor.

Length is positive, so one length is 6 in and the other length is 8 in.

048x2x2 2

Example 8 A right triangle has one leg that is 3 inches longer than the other leg. The hypotenuse is 15 inches. Find the unknown lengths.

x15

a2 + b2 = c2

x + 3222

15)3x(x 2259x6xx

22

2259x6x22

225225 0216x6x2

2

0)9x)(12x(2

9xor12x

a

b

c

One leg is 9 in and the other leg is 12 in.

129186274363542

108

0108x3x22

–108

3 12 –9

0)9x(0)12x(02

Example 1 The length of the hypotenuse is 10 cm.

Example 2 The length of the other leg is 9 cm.

Example 3 The diagonal length is or about 22.36 in.Example 4 The leg is 6” long.

Example 5 The longest line possible would be about 29.15 inches long.

Example 6 The lengths of the triangle are 3, 4, and 5.Example 7 Length is positive, so one length is 6 in and the other length is 8 in.

Example 8 One leg is 9 in and the other leg is 12 in.

510

10-A9 Page 552-554 #10-25, 53-58.