Lesson 50 Geometric Mean

Vocabulary New and Review The altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side, each triangle has 3 altitudes Hypotenuse is the side opposite the right angle in a right triangle Leg of a right triangle is one of the two sides that form the right angle In the proportion 𝑎 𝑏 = 𝑐 𝑑 , 𝑎 and 𝑑 are the extremes, and 𝑏 and 𝑐 are the means The geometric mean for positive numbers 𝑎 and 𝑑, is the positive number 𝑥 such that 𝑎 𝑥 = 𝑥 𝑑 .

Geometric Mean Find the geometric mean of 2 & 9 to the nearest tenth 2 𝑥 = 𝑥 9 𝑥 2 =18 𝑥≈4.2 Find the geometric mean of 5 & 11 to the nearest tenth 5 𝑥 = 𝑥 11 𝑥 2 =55 𝑥≈7.4

Geometric Mean 8 is the geometric mean of 16 & what number? 𝑎 8 = 8 16 16𝑎=64 𝑎=4 6 is the geometric mean of 3 & what number? 3 6 = 6 𝑑 3𝑑=36 𝑑=12

Theorem 50-1 If the altitude is drawn to the hypotenuse of a right tringle, then the two triangles formed are similar to each other and the original triangle. ∆𝐽𝑀𝐾~∆𝐾𝑀𝐿~∆𝐽𝐾𝐿

Corollary 50-1-1 If the altitude is drawn to the hypotenuse of a right triangle, then the length of the altitude is the geometric mean between the segments of the hypotenuse. 𝑎 𝑥 = 𝑥 𝑏

Corollary 50-1-2 If the altitude is drawn to the hypotenuse of a right triangle, then the length of the leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is closer to that leg. 𝑎 𝑥 = 𝑥 (𝑎+𝑏) or 𝑏 𝑦 = 𝑦 (𝑏+𝑎)

Review of Corollaries 50-1-1 & 50-1-2 Altitude is the Geometric Mean Leg is the Geometric Mean

Given ∆𝑆𝑇𝑄, find 𝑅𝑇 Altitude or Leg as the geo. mean? Altitude, Corollary 50-1-1 8 3 𝑥 = 𝑥 6 𝑥 2 =16 𝑥=4

Given the triangle, find 𝑐 and 𝑑 to the tenth Altitude or Leg as the geo. mean? Leg, Corollary 50-1-2 What is 𝑆𝑄, the hypotenuse? 𝑆𝑄=15, 3-4-5 factor of 3 𝑐 12 = 12 15 15𝑐=144

Given the triangle, find 𝑐 and 𝑑 to the tenth 𝑐= 144 15 𝑐=9.6 𝑑 9 = 9 15 15𝑑=81 𝑑= 81 15 𝑑=5.4

Looking Forward Finding the geometric mean and applying it to right triangles will prepare you for: Lesson 53: 45°-45°-90° Right Triangles Lesson 56: 30°-60°-90° Right Triangles Lesson 63: Introduction to Vectors Lesson 68: Introduction to Trigonometric Functions