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8-2 Special Right Triangles 45˚-45˚-90˚and 30˚-60˚-90˚ 45˚-45˚-90˚ Triangle • Two angles of a particular right triangle are 45˚ • What else can we say about this triangle, besides that it is a right triangle? • If each leg of the triangle has a measure of x length, what is the measure of the hypotenuse? 45˚-45˚-90˚ Triangle Theorem • We have just proved the 45˚-45˚-90˚ Triangle Theorem: • In a 45˚-45˚-90˚ triangle, both legs are congruent and the length of the hypotenuse is 2 times longer than the length of a leg. ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 2 ∙ 𝑙𝑒𝑔 Find the value of each variable • A. • B. More Practice: • Find the length of the hypotenuse of a 45˚-45˚-90˚ triangle with legs of length 5 3. • Find the length of a leg of a 45˚-45˚-90˚ triangle with a hypotenuse of length 10. • A square garden has sides 100ft long. You want to build a brick path along a diagonal of the square. To the nearest foot, how long will the path be? 30˚-60˚-90˚ Triangle Theorem • A similar relationship exists within right triangles with angle measures of 30˚ and 60˚. • The 30˚-60˚-90˚ Triangle Theorem says: • In a 30˚-60˚-90˚ triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is 3 times longer than the length of the shorter leg. ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 2 ∙ 𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔 𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔 = 3 ∙ 𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔 30˚-60˚-90˚ Triangle Theorem: Proof • ∆𝑊𝑋𝑌 is a 30˚-60˚-90˚ triangle within equilateral ∆𝑊𝑋𝑍. • If 𝑋𝑌 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑠, what is the length of 𝑊𝑋? • Use the Pythagorean theorem to solve for the length of 𝑊𝑋. Using 30˚-60˚-90˚ Triangle Theorem • Find the value of each variable. More Practice • In quadrilateral ABCD, AD = DC and AC = 20. Find the area of ABCD. Leave your answer in simplest radical form.