The Pythagorean Theorem and Its Converse LESSON 8-1 Additional Examples

A right triangle has legs of length 16 and 30. Find the length of the hypotenuse. Do the lengths of the sides form a Pythagorean triple?

a

2 +

b

2 =

c

2 16 2 + 30 2 =

c

2 256 + 900 =

c

2 1156 =

c

2 Use the Pythagorean Theorem.

Substitute 16 for

a

and 30 for

b

.

Simplify.

34 =

c

Take the square root.

The length of the hypotenuse is 34. The lengths of the sides, 16, 30, and 34, form a Pythagorean triple because they are whole numbers that satisfy

a

2 +

b

2 =

c

2 . Notice that each length is twice the common Pythagorean triple of 8, 15, and 17.

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The Pythagorean Theorem and Its Converse LESSON 8-1 Additional Examples

form.

Find the value of

x

. Leave your answer in simplest radical

a

2 +

b

2 =

c

2

x

2 + 10 2 = 12 2

x

2 + 100 = 144

x

2 = 44

x

= 4(11)

x

= 2 11

HELP

Use the Pythagorean Theorem.

Substitute

x

for

a

, 10 for

b

, and 12 for

c

.

Simplify.

Subtract 100 from each side.

Take the square root of each side.

Simplify.

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The Pythagorean Theorem and Its Converse LESSON 8-1 Additional Examples

A baseball diamond is a square with 90-ft sides. Home plate and second base are at opposite vertices of the square. About how far is home plate from second base?

Use the information to draw a baseball diamond.

a

2 +

b

2 =

c

2 90 2 + 90 2 =

c

2 8100 + 8100 =

c

2 16,200 =

c

2 Use the Pythagorean Theorem.

Substitute 90 for

a

Simplify.

and for

b

.

c

= 16,200 Take the square root.

c

Use a calculator.

The distance to home plate from second base is about 127 ft.

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The Pythagorean Theorem and Its Converse LESSON 8-1 Additional Examples

Is this triangle a right triangle?

a

2

+

b

2

c

2

4

2

+ 6

2

7

2

Substitute 4 for

a,

6 for

b,

and 7 for

c.

16 + 36 49 Simplify.

52 ≠ 49 Because

a

2

+

b

2

≠

c

2

,

the triangle is not a right triangle.

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The Pythagorean Theorem and Its Converse LESSON 8-1 Additional Examples

The numbers represent the lengths of the sides of a triangle. Classify each triangle as acute, obtuse, or right.

a.15, 20, 25 c

2

25

2

a

2

+

b

2

Compare

c

2

with

a

2

+

b

2

.

15

2

+ 20

2

Substitute the greatest length for

c.

625 225 + 400 Simplify.

625 = 625 Because

c

2

=

a

2

+

b

2

, the triangle is a right triangle.

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The Pythagorean Theorem and Its Converse LESSON 8-1 Additional Examples (Continued)

b. 10, 15, 20

c

2

a

2

+

b

2

20

2

10

2

+ 15

2

Compare

c

2

400 100 + 225 Simplify.

400 325 with

a

2

+

b

2

.

Substitute the greatest length for

c.

Because

c

2

a

2

+

b

2

, the triangle is obtuse.

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