Special Right Triangles Notes Part 1) Isosceles Right Triangles In this investigation you will simplify radicals to discover the relationship between the length of the legs and the length of the hypotenuse in a 45°-­‐45°-­‐90° triangle. Step 1) Find the length of the hypotenuse of each isosceles right triangle at right. Write each answer in simplest radical form. Step 2) Complete the table below. Draw triangles as needed. Length of each leg 1 2 3 4 5 6 7 10 15 𝑥 Length of the hypotenuse Step 3) Identify the pattern between the length of the legs and the length of the hypotenuse and use it to complete the conjecture below. Isosceles Right Triangle Conjecture In an isosceles right triangle, if the legs have length 𝑥 , then the hypotenuse has length_________. 45-­‐‑45-­‐‑90 Examples: Solve for each variable. Give your answers in simplest radical form. 1. 2. 3. 5√14 Part 2) 𝟑𝟎°-­‐𝟔𝟎°-­‐𝟗𝟎° Triangles In this investigation you will simplify radicals to discover the relationship between the length of the legs and the length of the hypotenuse in a 30°-­‐60°-­‐90° triangle. Step 1) Use the relationship between a 30°-­‐60°-­‐90° and an equilateral triangle to find the length of each hypotenuse in the triangles below. Then use the Pythagorean Theorem to calculate the length of the third side. Write your answer in simplified radical form. Step 2) Complete the table below. Draw triangles as needed. Length of the 1 2 3 4 5 6 7 10 15 𝑥 shorter leg Length of the hypotenuse Length of the longer leg Step 3) Identify the pattern between the length of the three sides of a 30°-­‐60°-­‐90° triangle to complete the conjecture below. 𝟑𝟎°-­‐𝟔𝟎°-­‐𝟗𝟎° Triangle Conjecture In a 30°-­‐60°-­‐90° triangle, if the shorter leg has length 𝑥 , then the longer leg has length _____ and the hypotenuse has length _____. 30-­‐‑60-­‐‑90 Examples: Solve for x and y . Give your answers in simplest radical form. 1. 2. 3.