Section 10.3 30˚-60˚-90˚ Triangles

Goal Find the side lengths of 30˚-60˚-90˚ triangles.

Key Vocabulary 30˚-60˚-90˚ triangles

Investigation The second special right triangle is the 30˚-60˚-90˚ right triangle, which is half of an equilateral triangle. Let’s start by using a little deductive reasoning to reveal a useful relationship in 30˚-60˚-90˚ right triangles.

Investigation Triangle ABC is equilateral, and segment CD is an altitude. What are m∠A and m∠B? What are m∠ADC and m∠BDC? What are m∠ACD and m∠BCD? Is ΔADC ≅ ΔBDC? Why? Is AD=BD? Why?

Investigation Notice that altitude CD divides the equilateral triangle into two right triangles with acute angles that measure 30° and 60°. Look at just one of the 30˚-60˚-90˚ right triangles. How do AC and AD compare? Conjecture: In a 30°-60°-90° right triangle, if the side opposite the 30° angle has length x, then the hypotenuse has length -?-. 2x

Investigation Find the length of the indicated side in each right triangle by using the conjecture you just made. 17 18 33 10 26 8.5

Investigation Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side. 6√3 4√3 5√3 8 6 10 8 50√3 4 50

Investigation You should have notice a pattern in your answers. Combine your observations with you latest conjecture and state your next conjecture. In a 30˚-60˚-90˚ triangle: short side = x hypotenuse = 2x long side = x√3

Theorem 10.2 30˚-60˚-90˚ Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse ℎ is 2 times the length of the shorter leg x, and the length of the longer leg is x√3 times the length of the shorter leg. Example:

30°-60°-90° Special Right Triangle In a triangle 30°-60°-90° , the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. Example: Hypotenuse 30° 2x Longer Leg 30° 10 cm x 5 cm 60° 60° x Shorter Leg 5 cm

Find the value of a and b. b = 14 cm 60° 7 cm 30 ° a = cm 30° 2x b 30 ° 60° a = cm a x Step 1: Find the missing angle measure. 30° Step 2: Match the 30°-60°-90° pattern with the problem. Step 3: From the pattern, we know that x = 7 , b = 2x, and a = x . Step 4: Solve for a and b

You can use the Pythagorean Theorem to find the value of b. Example 1 Find Leg Length In the diagram, PQR is a 30° –60° –90° triangle with PQ = 2 and PR = 1. Find the value of b. SOLUTION You can use the Pythagorean Theorem to find the value of b. (leg)2 + (leg)2 = (hypotenuse)2 Write the Pythagorean Theorem. 12 + b2 = 22 Substitute. 1 + b2 = 4 Simplify. b2 = 3 Subtract 1 from each side. Take the square root of each side. b = 3 13

hypotenuse = 2 · shorter leg Example 2 Find Hypotenuse Length In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the hypotenuse. SOLUTION The hypotenuse of a 30° –60° –90° triangle is twice as long as the shorter leg. hypotenuse = 2 · shorter leg 30° –60° –90° Triangle Theorem = 2 · 12 Substitute. = 24 Simplify. ANSWER The length of the hypotenuse is 24. 14

longer leg = shorter leg · 3 Example 3 Find Longer Leg Length In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the longer leg. SOLUTION The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times . 3 30° –60° –90° Triangle Theorem longer leg = shorter leg · 3 Substitute. = 5 · 3 ANSWER The length of the longer leg is 5 . 3 15

Your Turn: Find the value of x. Write your answer in radical form. 1. ANSWER 14 2. ANSWER 3 3. ANSWER 10 3

longer leg = shorter leg · 3 Example 4 Find Shorter Leg Length In the 30° –60° –90° triangle at the right, the length of the longer leg is given. Find the length x of the shorter leg. Round your answer to the nearest tenth. SOLUTION The length of the longer leg of a 30° –60° –90° triangle is the length of the shorter leg times 3 . 30° –60° –90° Triangle Theorem longer leg = shorter leg · 3 Substitute. 5 = x · 3 = x 5 3 Divide each side by . 2.9 ≈ x Use a calculator. ANSWER The length of the shorter leg is about 2.9. 17

hypotenuse = 2 · shorter leg Longer leg longer leg = shorter leg · 3 Example 5 Find Leg Lengths In the 30° –60° –90° triangle at the right, the length of the hypotenuse is given. Find the length x of the shorter leg and the length y of the longer leg. SOLUTION Use the 30° –60° –90° Triangle Theorem to find the length of the shorter leg. Then use that value to find the length of the longer leg. Shorter leg hypotenuse = 2 · shorter leg Longer leg longer leg = shorter leg · 3 8 = 2 · x y = 4 · 3 4 = x y = 4 3 18

The length of the shorter leg is x = 4. Example 5 Find Leg Lengths ANSWER The length of the shorter leg is x = 4. The length of the longer leg is y = 4 . 3 19

Your Turn: Find the value of each variable. Round your answer to the nearest tenth. 1. ANSWER 3.5 2. ANSWER x = 21; y = 21 ≈ 36.4 3

Your Turn: Find BC. A. 4 in. B. 8 in. C. D. 12 in.

Assignment Pg. 552 - 555 #1 – 49odd