Special Right Triangles Objectives: To use the properties of 45-45-90 and 30-60-90 right triangles to solve problems

Investigation 1 In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle. This triangle is also referred to as a 45-45-90 right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles.

Investigation 1 Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.

Investigation 1 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?

Special Right Triangle Theorem 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is times as long as each leg.

Example 1 Use deductive reasoning to verify the Isosceles Right Triangle Conjecture.

Example 2 A fence around a square garden has a perimeter of 48 feet. Find the approximate length of the diagonal of this square garden.

Investigation 2 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 2 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 2 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 -?-.

Investigation 2 Find the length of the indicated side in each right triangle by using the conjecture you just made.

Investigation 2 Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.

Investigation 2 You should have notice a pattern in your answers. Combine your observations with you latest conjecture and state your next conjecture.

Special Right Triangle Theorem 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

Two Special Right Triangles

Example 3 Find the value of each variable. Write your answer in simplest radical form.