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Geometry Lesson 8 – 1 Geometric Mean Objective: Find the geometric mean between two numbers. Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse. Geometric Mean Geometric mean The positive square root of their product. a x 2 , so x ab and x ab x b Geometric mean between 9 and 4. 9 x 2 , so x 36 and x 36 x 4 6 Find the geometric mean between 8 & 10. x ab x 8 10 Simplify the radical! x 24 25 What is the square root of 16? x4 5 Find the geometric mean 5 & 45 x ab x 5 45 x 559 Square root of 25? Square root of 9? x = 5(3) x = 15 12 & 15 x ab x 12 15 x 3 4 3 5 x 3(2) 5 x6 5 Theorem Theorem 8.1 If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Write a similarity statement identifying the three similar right triangles in the figure. The three triangles: FJG ~ GJH ~ FGH Write a similarity statement identifying the three similar right triangles in the figure. KML ~ MPL ~ KPM STR ~ QTS ~ QSR Theorem Geometric Mean (Altitude) Theorem The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments Theorem Geometric Mean (Leg) Theorem The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the the length of the the hypotenuse and the segment of the hypotenuse adjacent to that leg. Find x, y, and z. 20 x 25 y x 5 y 5 x 100 y 125 x = 10 25 z (leg theorem) z 20 z 500 10 5 22.4 5 5 11.2 Find x, y, & z 8 z z 25 z 200 z 10 2 14.1 33 y y 25 y 825 y 5 33 28.7 33 x x 8 x 264 x 2 66 16.2 Find x, y, and z. x 12 12 9 9x = 144 x = 16 16 25 z z 9 25 y y 16 z 225 y 400 z = 15 y = 20 Zach wants to order a banner that will hang over the side of his high school baseball stadium grandstand and reach the ground. To find this height, he uses a cardboard square to line up the top and bottom of the grandstand. He measures his distance from the grandstand and from the ground to his eye level. Find the height of the grandstand to the nearest foot. 5.75 10.5 10.5 x 5.75x = 110.25 x 19.17 Grandstand = 19.17 + 5 The grandstand is about 25 feet tall. Homework Pg. 535 1 – 7 all, 8 – 24 E, 28 – 36 EOE, 50, 54 – 74 EOE