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3 7-1 Objectives You will learn to: Find the geometric mean between two numbers. Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse.
4 Geometric Mean The geometric mean between two numbers is the positive square root of their product. For two positive numbers a and b, the geometric mean is the positive number x where the proportion a:x = x:b is true. This proportion can be written using fractions as = or with cross products as x 2 = ab or x = ab. a x x b
5 Find the geometric mean between 2 and 50. Let x represent the geometric mean. Definition of geometric mean Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is 10.
6 Find the geometric mean between 25 and 7. Let x represent the geometric mean. Definition of geometric mean Cross products Take the positive square root of each side. Simplify. Use a calculator. Answer: The geometric mean is about 13.2.
7 a. Find the geometric mean between 3 and 12. Answer: 6 b. Find the geometric mean between 4 and 20. Answer: 8.9
8 Similar Triangles Theorem 7.1 If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other. Y Example: ΔXYZ ~ ΔXWY ~ ΔYWZ X W Z
9 Altitude Geometric Mean Theorem 7.2 If the measure of an altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the altitude is the geometric mean between the measures of the two segments of the hypotenuse. Y X W Z Example: YW is the geometric mean of XW and ZW.
10 Cross products Take the positive square root of each side. Answer: Use a calculator. CD is about CD 2 = BD DA = 6 27 = 162 CD =
11 4 18 Answer: about 8.5
12 KITES Ms. Alspach is constructing a kite for her son. She has to arrange perpendicularly two support rods, the shorter of which is 27 inches long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two right triangles with the kite fabric, what is the length of the long rod?
13 Draw a diagram of one of the right triangles formed. Let be the altitude drawn from the right angle of XY 2 = WX ZX = 7.25 ZX = 7.25ZX ZX 25.1 Answer: The length of the long rod is , or about 32.4 inches long.
14 AIRPLANES A jetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, intersects at point E. If AE is 163 feet, what is the length of the aircraft? Answer: about ft
15 Hypotenuse Leg Theorem 7.3 If the altitude is drawn from the vertex of the right angle of the right triangle to its hypotenuse, then the measure of a leg of the triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to the leg. Example: Y XZ XY XY XZ = and = XW YZ YZ WZ X W Z
16 Find c and d in is the altitude of right triangle JKL. Use Theorem 7.2 to write a proportion. is the leg of right triangle JKL. Use the Theorem 7.3 to write a proportion. Answer: c 20 and d 11.2
17 Find e and f. f Answer:
18 What did you learn today? How to: Find the geometric mean between two numbers. Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse.
19 Assignment: Page odd, 41, 56, 60
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