Sec. 7 – 4 Similarity in Right Δ: Sec. 7 – 4 Similarity in Right Δ Objectives: 1) To find & use relationships in similar right Δ .
Geometric Mean: Geometric Mean In the proportion b & c are the means. Whenever the means are equal, it is a Meaning b = c They can be replaced with the same variable a b = c d a x = x d Geometric Mean Geometric Mean Extreme Extreme Geometric Mean
Ex1: Find the geometric mean between 2 numbers.: Ex1: Find the geometric mean between 2 numbers. 4 & 18 15 & 20 4 x = x 18 Extreme Extreme Step 2: Find the cross Products Step 1: Set up a Proportion x 2 = 72 x = √72 x = √36•2 x = 6√2 = 8.5 Geometric Mean Geometric Mean 15 x = x 20 x 2 = 300 x = √300 x = √100•3 x = 10√3 = 17.3
Ex.2: 8 is the geometric mean between 17 and what number?: Ex.2: 8 is the geometric mean between 17 and what number? = 8 8 17 x 17x = 64 X = 3.8
Slide 5: 7-3
* In a right Δ, the altitude to the hypotenuse yields 3 similar Δs.: * In a right Δ , the altitude to the hypotenuse yields 3 similar Δ s. Th.(7 – 3) The altitude to the hypotenuse of a right Δ divides the Δ into 2 Δ s that are similar to the original Δ & to each other. B C A 1 2 3 4 5 6 Δ ABC ~ D Δ ACD ~ Δ CBD 1 2 3 5 6 4 C AC ~ AD ~ DC ~ CB ~ CD ~ DB ~ AB AC CB This page is the same as the last page but with all the similarity statements
Slide 7: Example 1: Identifying Similar Right Triangles Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. By Theorem 7-3, ∆ UVW ~ ∆ UWZ ~ ∆ WVZ . Z W
Altitude Corollary: Altitude Corollary The length of the altitude to the hypotenuse of a right Δ is the geometric mean of the lengths of the segment of the hypotenuse. The altitude splits the hyp. Into 2 segments. = x y h h
Ex3: Find the missing variables : Ex3: Find the missing variables B A D C y x 6 2 2x = 36 x = 18 a 2 + b 2 = c 2 6 2 + 18 2 = c 2 36 + 324 = c 2 c = 18.9 = 6 6 x 2
Leg Corollary: Leg Corollary The altitude to the hyp. of a right Δ seperates the hyp. so that the length of each leg of a Δ is the geometric mean between the length of the adjacent part of the hyp. and the length of the entire hyp. C = x c a a = y c b b
Leg Corollary: Leg Corollary The altitude to the hyp. of a right Δ seperates the hyp. so that the length of each leg of a Δ is the geometric mean between the length of the adjacent part of the hyp. and the length of the entire hyp. C = x c a a = y c b b
Leg Corollary: Leg Corollary The altitude to the hyp. of a right Δ seperates the hyp. so that the length of each leg of a Δ is the geometric mean between the length of the adjacent part of the hyp. and the length of the entire hyp. C = y c b b = x c a a
Ex.4 Solve for x & y: Ex.4 Solve for x & y x y 4 5 y 2 = 20 y = √ 20 y = 4.5 x 2 = 36 x = √ 36 x = 6 = x x 9 4 = y y 5 4
Ex.5: Might have to use Pythagoras first!: Ex.5: Might have to use Pythagoras first! 3 y x z 4 a 2 + b 2 = c 2 3 2 + 4 2 = c 2 c = 5 = 3 3 5 x x = 1.8 z = 5 – 1.8 z = 3.2 = y y 3.2 1.8 y = 2.4
What’s New???: What’s New??? = Part of Hyp Part of hyp Altitude Altitude Altitude Corollary = Adj Part of Hyp Whole hyp Leg Leg Leg Corollary Geometric Mean