7.4 Similarity in Right Triangles

In this lesson we will learn the relationship between different parts of a right triangle that has an altitude drawn in it

Geometric Mean

Geometric Mean: The number x such that , where a, b, and x are positive numbers

If we solve we get x2=ab, so

a

xx

b

abx

Before we look at right triangles we will examine something called the GEOMETRIC MEAN

You could solve the proportion OR take the short cut

x2=36 x=6 x=6

4

9 x

x 49x

36x

Ex. Find the geometric mean between 9 and 4.

Geometric Mean

Geometric Mean: The number x such that , where a, b, and x are positive numbers

If we solve we get x2=ab, so

a

xx

b

abx

You could solve the proportion OR take the short cut

x2=150

15

10 x

x

65x

1510x150x

65x

Ex. Find the geometric mean between 10 and 15.

Practice Problems

Geometric Mean: The number x such that , where a, b, and x are positive numbers

If we solve we get x2=ab, so

a

xx

b

abx

Put these two problems on your direction sheet1.Find the geometric mean between 5 and 202.Find the geometric mean between 12 and 15.

Similarity in Right Triangles

Theorem 7-3: The altitude to the hypotenuse of a right triangle divides the triangles into two triangles that are similar to the original triangle and to each other.

Geometric Mean with Altitude

5.2 in 8.75in

6.75in

Corollary to Theorem 7-3: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse

75.82.575.6

75.8

75.6

75.6

2.5

So, since 6.75 is the altitude, it is the geometric mean of 5.2 and 8.75

Similarity in Right TrianglesEx. Find the values of x in the following right triangles.

9 7

3

73x

63x79x

5

x

x

x325 x35

x3/25

x is the geometric mean of 9 and 7

5 is the geometric mean of x and 3

Practice ProblemsPut these three problems on your direction sheet. Find y in each picture.

3.

28

y4.

19

y

9

5.

Geometric MeanSecond Corollary to Theorem 7-3: The altitude to the hypotenuse separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the lengths of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg.

6

6

6 is the geometric mean of 3 and 123 is the part of the hypotenuse closest to side of 6. 12 is the whole hypotenuse

31236

12

6

6

3

Geometric Meanf is the geometric mean of 10 and 12

2f

10Example.

1210f

120f

302f

2 7

w

w is the geometric mean of 2 and 9

18w

92w

23w

Practice Problems

7. Find w, j

A

C

D B

w

54

jA

CDB

w

12

8

j

8. Find w, j

Put these two problems on your direction sheet

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