ExerciseExerciseSolve x2 = 4.Solve x2 = 4.

x = ± 2x = ± 2

Solve x2 = – 4.Solve x2 = – 4.

no real solutionno real solution

ExerciseExercise

Solve √ x = 4.Solve √ x = 4.

x = 16x = 16

ExerciseExercise

Solve √ x = – 4.Solve √ x = – 4.

no real solutionno real solution

ExerciseExercise

Solve √ – x = 4.Solve √ – x = 4.

x = – 16x = – 16

ExerciseExercise

legleg

legleghypotenuse

hypotenuse

55 3344

9 square units

9 square units

25 square units

25 square units

16 square units

16 square units

55 3344

9 square units

9 square units

25 square units

25 square units

16 square units

16 square units

+ =+ =

+ =+ =16 9 2516 9 2542 32 5242 32 52

The Pythagorean TheoremThe Pythagorean Theorem

If the hypotenuse of a right triangle has length c, and the

legs have lengths a and b, then a2 + b2 = c2.

If the hypotenuse of a right triangle has length c, and the

legs have lengths a and b, then a2 + b2 = c2.

Find the hypotenuse of a right triangle with legs of 8 and 15.

Find the hypotenuse of a right triangle with legs of 8 and 15.

c = 17c = 17

c2 = a2 + b2c2 = a2 + b2

c2 = 82 + 152c2 = 82 + 152

c2 = 64 + 225c2 = 64 + 225

c2 = 289c2 = 289√ c2 = √ 289√ c2 = √ 289

Example 1Example 1

Find the hypotenuse of a right triangle with legs of 6 and 7.

Find the hypotenuse of a right triangle with legs of 6 and 7.c2 = a2 + b2c2 = a2 + b2

c2 = 62 + 72c2 = 62 + 72

c2 = 36 + 49c2 = 36 + 49

c2 = 85c2 = 85√ c2 = √ 85√ c2 = √ 85

c = √ 85 ≈ 9.2c = √ 85 ≈ 9.2

Example 2Example 2

Find the hypotenuse of a right triangle with legs of 9 and 12.

Find the hypotenuse of a right triangle with legs of 9 and 12.

1515

ExampleExample

Find the hypotenuse of a right triangle with legs of

and .

Find the hypotenuse of a right triangle with legs of

and .

11

1212

√ 32

√ 32

ExampleExample

Find the hypotenuse of a right triangle with legs of 1 and 1.

Find the hypotenuse of a right triangle with legs of 1 and 1.

√ 2√ 2

ExampleExample

Find the leg of a right triangle whose hypotenuse is 16 and other leg is 7.

Find the leg of a right triangle whose hypotenuse is 16 and other leg is 7.a2 + 72 = 162a2 + 72 = 162 a2 + 49 = 256a2 + 49 = 256

a2 = 207a2 = 207a = √ 207 ≈ 14.4a = √ 207 ≈ 14.4

a2 + 49 – 49 = 256 – 49a2 + 49 – 49 = 256 – 49

Example 3Example 3

Find the length of a leg of a right triangle whose hypotenuse is 39 and whose other leg is 15.

Find the length of a leg of a right triangle whose hypotenuse is 39 and whose other leg is 15.

3636

ExampleExample

Find the length of a leg of a right triangle whose hypotenuse is 20 and whose other leg is 10.

Find the length of a leg of a right triangle whose hypotenuse is 20 and whose other leg is 10.

√ 300 ≈ 17.3√ 300 ≈ 17.3

ExampleExample

The converse is the statement resulting when the “if” part and the “then” part of a conditional statement are switched.

The converse is the statement resulting when the “if” part and the “then” part of a conditional statement are switched.

ConverseConverse

Converse of the Pythagorean Theorem

Converse of the Pythagorean Theorem

If a triangle has sides a, b, and c, such that

a2 + b2 = c2, then the triangle is a right triangle.

If a triangle has sides a, b, and c, such that

a2 + b2 = c2, then the triangle is a right triangle.

Determine whether a triangle with sides of 12, 35, and 37 is a right triangle.

Determine whether a triangle with sides of 12, 35, and 37 is a right triangle.

1,369 = 1,3691,369 = 1,369

a2 + b2 = c2a2 + b2 = c2

122 + 352 = 372122 + 352 = 372

144 + 1,225 = 1,369144 + 1,225 = 1,369

Example 4Example 4

yesyes

Determine whether a triangle with sides of 8, 12, and 14 is a right triangle.

Determine whether a triangle with sides of 8, 12, and 14 is a right triangle.

208 ≠ 196208 ≠ 196

a2 + b2 = c2a2 + b2 = c2

82 + 122 = 14282 + 122 = 142

64 + 144 = 19664 + 144 = 196

Example 5Example 5

nono

Determine whether a triangle with sides of 15, 18, and 22 is a right triangle.

Determine whether a triangle with sides of 15, 18, and 22 is a right triangle.

no; 152 + 182 ≠ 222no; 152 + 182 ≠ 222

ExampleExample

Determine whether a triangle with sides of 16, 30, and 34 is a right triangle.

Determine whether a triangle with sides of 16, 30, and 34 is a right triangle.

yes; 162 + 302 = 1,156 = 342yes; 162 + 302 = 1,156 = 342

ExampleExample

A 16 ft. ladder leans up against the side of a building. If the base of the ladder is 4 ft. from the base of the building, how high up the side of the building will the ladder reach?

A 16 ft. ladder leans up against the side of a building. If the base of the ladder is 4 ft. from the base of the building, how high up the side of the building will the ladder reach?

15.5 ft.15.5 ft.

ExerciseExercise

A 200 ft. tower is braced to the ground by a cable, from a point 150 ft. above the ground to a point 87 ft. from the base of the tower. How long is the cable?

A 200 ft. tower is braced to the ground by a cable, from a point 150 ft. above the ground to a point 87 ft. from the base of the tower. How long is the cable?

173.4 ft.173.4 ft.

ExerciseExercise

The distance between bases on a baseball diamond is 90 ft. How far is it from home plate to second base?

The distance between bases on a baseball diamond is 90 ft. How far is it from home plate to second base?

127.3 ft.127.3 ft.

ExerciseExercise

An opening for a window is 23” wide, 54” tall, and 60” diagonally. Is the opening “square”; that is, do the height and width form a right angle?

An opening for a window is 23” wide, 54” tall, and 60” diagonally. Is the opening “square”; that is, do the height and width form a right angle?

nono

ExerciseExercise