Similarity in Right Triangles

Students will be able to find segment lengths in right

triangles, and to apply similarity relationships

in right triangles to solve problems.

Unit F 2

Geometric Mean

• Remember that in a proportion such as , a and b are called the extremes and r and q are called the means.

• The geometric mean of two numbers is the positive square root of their product. We use the following proportion to find the geometric mean:

• Notice that the means both have x. That is the geometric mean. How do you solve for x?

→ →

a

q b

r

a

x b

x

a

x b

x x a b

2x a b

Unit F 3

Examples of Geometric Means

• Find the geometric mean between 4 and 9.

→ x2 = 36 → → x = 6

• Now, find the geometric mean between 2 and 8.

→ x2 = 16 → → x = 4

4

9

x

x 36x

2

8

x

x 16x

Unit F 4

Similar Right Triangles Theorem

• The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.

C

BDA

∆ABC ∼ ∆ACD ∼ ∆CBD

Unit F 5

• The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the two lengths of the segments of the hypotenuse.

• This means: The altitude is the geometric mean of the two segments of the hypotenuse

and .

• Or we could say:

CD2 = AD ∙ BD

C

BD

A

AD CD

CD BD

BDAD

CD

Geometric Means Corollary

Unit F 6

To estimate the height of a Douglas fir, Jan positions herself so that her lines of sight to the top and bottom of the tree form a 90º angle. Her eyes are about 1.6 m above the ground, and she is standing 7.8 m from the tree. What is the height of the tree to the nearest meter?

Example of Corollary

Set up the proportion:

1.6 7.8

7.8 x→ 1.6x = 7.8 ∙ 7.8 x

1.6x = 60.84 → x ≈ 38

So the tree is about 40 meters tall.