The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1-6, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

a2 + b2 = c2

Example 1A: Find the value of x. Give your answer in simplest radical form.

a2 + b2 = c2

22 + 62 = x2

40 = x2

Example 1B: Find the value of x. Give your answer in simplest radical form.

a2 + b2 = c2

(x – 2)2 + 42 = x2

x2 – 4x + 4 + 16 = x2

–4x + 20 = 0

20 = 4x

5 = x

Example 2:Randy is building a rectangular picture frame. He wants the ratio of the length to the width to be 3:1 and the diagonal to be 12 centimeters. How wide should the frame be? Round to the nearest tenth of a centimeter.

Let l and w be the length and width in centimeters of the picture. Then l:w = 3:1, so l = 3w.

a2 + b2 = c2

(3w)2 + w2 = 122

10w2 = 144

A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a Pythagorean triple.

Example 3A:Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

The side lengths are nonzero whole numbers that satisfy the equation a2 + b2 = c2, so they form a Pythagorean triple.

a2 + b2 = c2

142 + 482 = c2

2500 = c2

50 = c

Example 3B:

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

a2 + b2 = c2

42 + b2 = 122

b2 = 128

The side lengths do not form a Pythagorean triple because is not a whole number.

The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths.

You can also use side lengths to classify a triangle as acute or obtuse.

A

B

C

c

b

a

To understand why the Pythagorean inequalities are true, consider ∆ABC.

By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greaterthan the third side length.

Remember!

Example 4a Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

Step 1 Determine if the measures form a triangle.

7, 12, 16

7 + 12 > 16, 7 + 16 > 12, 12 + 16 > 7

so, 7, 12, and 16 can be the side lengths of a triangle.

Step 2 Classify the triangle.

Since c2 > a2 + b2, the triangle is obtuse.

256 > 193

c2 = a2 + b2?

162 = 122 + 72?

256 = 144 + 49?

Example 4b

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

Step 1 Determine if the measures form a triangle.

11, 18, 34

Since 11 + 18 = 29 and 29 > 34, these cannot be the sides of a triangle.