Bell Ringer

A right triangle with angles measure of 45,45, and 90 are called 45-45-90 triangles

By the 45° –45° –90° Triangle Theorem, the length of the Example 1 Find Hypotenuse Length Find the length x of the hypotenuse in the 45° –45° –90° triangle shown at the right. SOLUTION By the 45° –45° –90° Triangle Theorem, the length of the hypotenuse is the length of a leg times . 2 45° –45° –90° Triangle Theorem hypotenuse = leg · 2 Substitute. = 3 · 2 ANSWER The length of the hypotenuse is 3 . 2 3

Find the length x of each leg in the Example 2 Find the length x of each leg in the 45° –45° –90° triangle shown at the right. Find Leg Length SOLUTION By the 45° –45° –90° Triangle Theorem, the length of the hypotenuse is the length of a leg times . 2 45° –45° –90° Triangle Theorem hypotenuse = leg · 2 Substitute. = x 7 2 = x 2 7 Divide each side by . 7 = x Simplify. ANSWER The length of each leg is 7. 4

Now You Try  Find the value of x. 1. ANSWER 4 2 2. ANSWER 5 2 3. Find Hypotenuse and Leg Lengths Find the value of x. 1. ANSWER 4 2 2. ANSWER 5 2 3. ANSWER 3 ANSWER 6 4.

By the Triangle Sum Theorem, x° + x° + 90° = 180°. Example 3 Identify 45° –45° –90° Triangles Determine whether there is enough information to conclude that the triangle is a 45° –45° –90° triangle. Explain your reasoning. SOLUTION By the Triangle Sum Theorem, x° + x° + 90° = 180°. So, 2x° = 90°, and x = 45. ANSWER Since the measure of each acute angle is 45°, the triangle is a 45° –45° –90° triangle. 6

You can use the 45° –45° –90° Triangle Theorem to find the value of x. Example 4 Show that the triangle is a 45° –45° –90° triangle. Then find the value of x. Find Leg Length SOLUTION The triangle is an isosceles right triangle. By the Base Angles Theorem, its acute angles are congruent. From the result of Example 3, this triangle must be a 45° –45° –90° triangle. You can use the 45° –45° –90° Triangle Theorem to find the value of x. 45° –45° –90° Triangle Theorem · hypotenuse = leg 2 Substitute. = x 5 2 7

Use a calculator to approximate. Example 4 Find Leg Length Divide each side by . 2 = x 5 Simplify. = x 5 2 3.5 ≈ x Use a calculator to approximate. 8

Now You Try  Find Leg Lengths Show that the triangle is a 45° –45° –90° triangle. Then find the value of x. Round your answer to the nearest tenth. 5. ANSWER x = ≈ 5.7. The triangle is an isosceles right triangle. By the Base Angles Theorem, its acute angles are congruent. From the result of Example 3, the triangle is a 45° –45° –90° triangle. 8 2

Checkpoint Now You Try  Find Leg Lengths Show that the triangle is a 45° –45° –90° triangle. Then find the value of x. Round your answer to the nearest tenth. 6. ANSWER The triangle has congruent acute angles. By Example 3, the triangle is a 45° –45° –90° triangle. x = ≈ 8.5. 12 2