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Lesson 7.3 Two Special Right Triangles Objectives: To use properties of 45-45-90 triangles To use properties of 30-60-90 triangles Mrs. McConaughy Geometry 1 Isosceles Right Triangle Theorem ISOSCELES RIGHT TRIANGLE NOTE: If you are given the length of THEOREM:theInhypotenuse, an isosceles you right can determine the triangle, if the legs have length, l, length of a side by dividing l it√2 then the hypotenuse has length ____. √2, then rationalizing the by_________________________ denominator, when necessary. ___________________________. Mrs. McConaughy Geometry 2 EXAMPLES: Find the length of the hypotenuse in each isosceles triangle below. 3√2 6√2 Mrs. McConaughy 4√2 5√2 7√2 12√2 Geometry 3 Recall: Triangle Inequalities If two angles of a triangle are not congruent, then the longest side largest angle lies opposite the _______ and the shortest side lies opposite smallest the ________ angle. Mrs. McConaughy Geometry 4 30-60-90 TRIANGLE THEOREM 30-60-90 TRIANGLE THEOREM: In a 30-6090 triangle, if the side opposite the 30 degree angle has length, l, the hypotenuse has length 2l _______. NOTE: These triangles are sometimes referred to as 1-2-√3 right triangles. Mrs. McConaughy Geometry 5 Easy way to remember the relationship among angles and sides in 30-60-90 triangles: 1. Rank order the following numbers from smallest to largest: 1, 2, √3 2l 60 1, √3 , 2 2. Now, use the Triangle Inequality Theorem to place the side lengths 1l, √3l , 2l opposite the appropriate angles in a 30-60-90 triangle. Mrs. McConaughy 30 l√3 NOTE: It is usually easier to determine the length of the shortest and longest sides, Geometry 6 initially. 1l Find the length of each indicated side: 60 ____ ____ ____ 30 NOTE: The length of one side will be provided Mrs. McConaughy Geometry 7 by your instructor. Find the length of each indicated side. Mrs. McConaughy Geometry 8 In summary: We can find the lengths of sides in right triangles by using: 30-60-90 ∆ Pythagorean Primitives c a …and ∆ 3 45-45-90 •4•5 5 • 12 • 13 l their ! 2l 8 • 15 multiples • 17 7 • 24 • 25 45 b l l√2 45 Pythagorean Theorem c = a2 + b 2 Mrs. McConaughy 2 Geometry 30 l√3 60 l 9 Putting it all together: Find the length of each indicated side. 20√3 8 ∙ 3 8 ∙5 __ =40 __ 20 8 ∙ 4 Mrs. McConaughy Geometry 10 Homework Assignment: Special Right Triangles WS (1-10 all, 12) Mrs. McConaughy Geometry 11