Post on 25-Dec-2015
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.