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Name: __________________________ Date: ______________ Pd: ______ 8-1 Geometric Mean and Pythagorean Theorem Triangle Inequality Review: _______________________________________________________ ______________________________________________________________________________ Converse of the Pythagorean Theorem REMEMBER: For RIGHT triangles:____________________________________ For ACUTE triangles:____________________________________ For OBTUSE triangles:___________________________________ Determine if the set of numbers can be the sides of a triangle. If yes, classify the triangle. 1. 4 cm, 5 cm, 3 cm 2. 6 in, 9 in, 7 in 3. 15 ft, 30 ft, 15 ft 4. 10 m, 10 m, 10 m 5. 8 yd, 12 yd, 8 yd 6. 50°, 50°, 80° 7. 20°, 90°, 60° 8. 150°, 20°, 10° Ratios & Proportions: Express the following as a ratio (fraction) in simplest form. 1. 24 to 42 2. 50 minutes to 2 hours 3. 80 centimeters to 3 meters 4. 10 weeks to 35 days A 5. AC AB 6. CD BD 7. AD BD 8. AC BC Find AC and AD first! ___ 10 D ____ 3.6 4.8 C 6 B To solve a proportion, ___________________________________________________________. 9. 2 6 3 x 13. x3 x3 2 5 10. 10 6 6 x 14. 3x 2 2x 7 4 3 11. 9 x x 16 15. 12. 8 x5 7 x 2 8x 12.8 2x 8 3 2.5 Geometric Mean Formula: Find the geometric mean of each pair of numbers. ________1. 4 and 9 ________2. 5 and 10 ________3. 3 and 15 Altitudes in Right Triangles When a right triangle has an altitude, the segment lengths have special ratios. RULE #1: The altitude is the geometric mean of the two segments of the hypotenuse. Altitude: B BD AD and DC Hypotenuse Segments: D A Formula: C AD BD (remember the formula for geometric mean!) BD DC Use the formula and triangle from above to solve the following problems. 10 x x 2 1. Suppose AD = 10 and DC = 2. Find BD. x2 = 20 x 20 4 5 2 5 2. Suppose BD = 10 and AD = 25. Find DC. _____ = _____ DC = ____________ 3. Using the answer for #2, find AC. (Hint: AC = ? + ?) AC = ____________ 4. Suppose AC = 12 and DC = 4. Find AD first. Then use the formula to find BD. AD = ____________ _____ = _____ BD = ____________ Write the altitude formula (geometric mean) for the following triangles. F 5. E 6. H G W M Z X J 7. Y L _____ = _____ _____ = _____ _____ = _____ K RULE #2: A leg of the right triangle is the geometric mean between the whole hypotenuse and the part of the hypotenuse adjacent to that leg. B Leg = AB Whole Hypotenuse: AC D A C Adjacent Part of Hypotenuse: AD Formula: AC AB AB AD Note: If BC is used instead of AB , then DC would replace AD . Use the formula and triangle from above to solve the following problems. 8. Suppose AC = 12 and DC = 2. Find AD first. Then find AB. 12 x x 120 4 30 2 30 AD = 12 – 2 = 10 x2 = 120 x 10 9. Suppose AB = 12 and AD = 9. Find AC. _____ = _____ AC = ____________ 10. Using the answer from #9, find DC. DC = ____________ 11. Suppose BC = 8 and DC = 4. Find AC first. Then find AD. _____ = _____ AC = ____________ AD = ____________ 12. Write the geometric mean formulas for each leg of ∆EFG. F Leg = EF Leg = FG _____ = _____ _____ = _____ E H G