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Exercise. Solve x 2 = 4. x = ± 2. Exercise. Solve x 2 = – 4. no real solution. Exercise. Solve √ x = 4. x = 16. Exercise. Solve √ x = – 4. no real solution. Exercise. Solve √ – x = 4. x = – 16. hypotenuse. leg. leg. 25 square units. 9 square units. 3. 5. 4.

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Exercise

Solve x2 = 4.

x = ± 2


Exercise

Solve x2 = –4.

no real solution


Exercise

Solve √x = 4.

x = 16


Exercise

Solve √x = –4.

no real solution


Exercise

Solve √–x = 4.

x = –16


hypotenuse

leg

leg


25 square units

9 square units

3

5

4

16 square units


3

5

4

25 square units

9 square units

+=

16 square units


42 32 52

+=

16 9 25


The Pythagorean Theorem

If the hypotenuse of a right triangle has length c, and the legs have lengths a and b, then a2 + b2 = c2.


c2 = √289

Example 1

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

c2 = a2 + b2

c2 = 289

c2 = 82 + 152

c2 = 64 + 225

c = 17


c2 = √85

c = √85 ≈ 9.2

Example 2

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

c2 = a2 + b2

c2 = 85

c2 = 62 + 72

c2 = 36 + 49


Example

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

15


Example

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

12

√32

1


2

Example

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


a = √207 ≈ 14.4

Example 3

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

a2 + 72 = 162

a2 + 49 = 256

a2 + 49 – 49 = 256 – 49

a2 = 207


Example

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

36


300 ≈ 17.3

Example

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


Converse

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


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.


Example 4

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

a2 + b2 = c2

yes

122 + 352 = 372

144 + 1,225 = 1,369

1,369 = 1,369


Example 5

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

a2 + b2 = c2

no

82 + 122 = 142

64 + 144 = 196

208 ≠ 196


Example

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

no; 152 + 182 ≠ 222


Example

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

yes; 162 + 302 = 1,156 = 342


Exercise

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.


Exercise

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.


Exercise

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

127.3 ft.


Exercise

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?

no


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