Section 10.2

Goal  Find the side lengths of 45 ˚ -45 ˚ -90 ˚ triangles.

Key Vocabulary  45 ˚ -45 ˚ -90 ˚ triangle  Isosceles right triangle  Leg of a right triangle  hypotenuse

Side Lengths of Special Right ∆s  Right triangles whose angle measures are 45° - 45° - 90° or 30° - 60° - 90° are called special right triangles.  Because of the special relationships between the side lengths of special right triangles we can solve them without using the pythagorean theorem.

Investigation This triangle is also referred to as a right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles. In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle.

Investigation Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.

Investigation Did you notice something interesting about the relationship between the length of the hypotenuse and the length of the legs in each problem of this investigation?

Theorem ˚ -45 ˚ -90 ˚ Triangle Theorem 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the legs ℓ are congruent and the length of the hypotenuse ℎ is times the length of a leg. Example:

45°-45°-90° Special Right Triangle  In a triangle 45°-45°-90°, the hypotenuse is times as long as a leg. 45° Hypotenuse x x x Leg Example: 45° 5 cm 5 cm

Find the value of a and b. 45° 7 cm a b Step 1: Find the missing angle measure. 45° Step 2: Match the 45°-45°-90° pattern with the problem. 45° x x x Step 3: From the pattern, we know that x = 7, a = x, and b = x. a = 7 cm b = 7 cm Step 4: Solve for a and b

Example 1 Find Hypotenuse Length 45° –45° –90° Triangle Theorem hypotenuse = leg · 2 ANSWER The length of the hypotenuse is 3. 2 Substitute. = 3 · 2 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

Example 2 Find Leg Length ANSWER The length of each leg is 7. Find the length x of each leg in the 45° –45° –90° triangle shown at the right. 7 = x Simplify.Substitute. = x722 45° –45° –90° Triangle Theorem hypotenuse = leg · 2 SOLUTION By the 45° –45° –90° Triangle Theorem, the length of the hypotenuse is the length of a leg times. 2 = x Divide each side by. 2

Your Turn: ANSWER 3 6 Find the value of x ANSWER

Your Turn: A. Find x. A.3.5 B.7 C. D.

Your Turn: B. Find x. A. B. C.16 D.32

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

Example 4 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. Show that the triangle is a 45° –45° –90° triangle. Then find the value of x. 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

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

Your Turn: Show that the triangle is a 45° –45° –90° triangle. Then find the value of x. Round your answer to the nearest tenth. 1. 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

Your Turn: Show that the triangle is a 45° –45° –90° triangle. Then find the value of x. Round your answer to the nearest tenth. 2. ANSWER The triangle has congruent acute angles. By Example 3, the triangle is a 45° –45° –90° triangle. x = ≈

Your Turn: Find b. A. B.3 C. D.

Cartoon Time

Assignment  Pg #1 – 27odd, 31, 35 – 45 odd