8-2 T

HE PYT

HAGOREAN

THEOREM A

ND ITS

CONVERSE

8-2

(For help, go to the Skills Handbook, page 753.)

1. 2.

3. 4.

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

Square the lengths of the sides of each triangle. What do you notice?

Check Skills You’ll Need

1. 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; 9 + 16 = 25

2. 52 = (5)(5) = 25; 122 = (12)(12) = 144; 132 = (13)(13) = 169; 25 + 144 = 169

3. 62 = (6)(6) = 36; 82 = (8)(8) = 64; 102 = (10)(10) = 100; 36 + 64 = 100

4. 42 = (4)(4) = 16; (4 2 )2 = (4 2)(4 2) = 16 (2)(2) = 16(2) = 32; 16 + 16 = 32.

Solutions

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

8-2

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?

a2 + b2 = c2 Use the Pythagorean Theorem.

162 + 302 = c2 Substitute 16 for a and 30 for b.

256 + 900 = c2 Simplify.

1156 = c2

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 a2 + b2 = c2. Notice that each length is twice the common Pythagorean triple of 8, 15, and 17.

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

Quick Check

8-2

a2 + b2 = c2 Use the Pythagorean Theorem.

x2 + 102 = 122 Substitute x for a, 10 for b, and 12 for c.

x2 + 100 = 144 Simplify.

x2 = 44 Subtract 100 from each side.

x = 4(11) Take the square root of each side.

x = 2 11 Simplify.

Find the value of x. Leave your answer in simplest radical form.

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

Quick Check

8-2

c = 16,200 Take the square root.

c 127.27922 Use a calculator.

a2 + b2 = c2 Use the Pythagorean Theorem.

902 + 902 = c2 Substitute 90 for a and for b.

8100 + 8100 = c2 Simplify.

16,200 = c2

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

Use the information to draw a baseball diamond.

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?

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

Quick Check

8-12

Is this triangle a right triangle?

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

52 ≠ 49

Because a2 + b2 ≠ c2, the triangle is not a right triangle.

Quick Check

8-2

a2 + b2 c2

42 + 62 72 Substitute 4 for a, 6 for b, and 7 for c.

16 + 36 49 Simplify.

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

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

a.15, 20, 25

c2 a2 + b2 Compare c2 with a2 + b2.

625 225 + 400 Simplify.

625 = 625

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

8-2

252 152 + 202 Substitute the greatest length for c.

(Continued)

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

b. 10, 15, 20

400 325

Because c2 a2 + b2, the triangle is obtuse.

Quick Check

8-2

202 102 + 152 Substitute the greatest length for c.

400 100 + 225 Simplify.

c2 a2 + b2 Compare c2 with a2 + b2.

2 33

1. Find the value of x.

2. Find the value of x. Leave your answer in simplest radical form.

3. The town of Elena is 24 mi north and 8 mi west of Holberg. A train runs on a straight track between the two towns. How many miles does it cover? Round your answer to the nearest whole number.

4. The lengths of the sides of a triangle are 5 cm, 8 cm, and 10 cm. Is it acute, right, or obtuse?

15

25 mi

obtuse

GEOMETRY LESSON 8-1GEOMETRY LESSON 8-1

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

8-2