Transcript Slide 1

Geometry B
Chapter 8
Geometric Mean
Your Math Goal Today is…
Use geometric mean to find segment
lengths in right triangles.
Apply similarity relationships in right
triangles to solve problems.
Warm Up
1. Write a similarity statement
comparing the two triangles.
∆ADB ~ ∆EDC
Simplify.
2.
3.
Solve each equation.
4.
5. 2x2 = 50 ±5
Vocabulary
geometric mean
In a right triangle, an altitude drawn from the
vertex of the right angle to the hypotenuse forms
two right triangles.
Please turn to page 538 Theorem 8.1
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.
W
Z
By Theorem 8.1, ∆UVW ~ ∆UWZ ~ ∆WVZ.
Note:
Writing similarity statements for triangles,
be sure to name the vertices in the corresponding
order in each triangle.
In Your Notes! Example 1
Write a similarity statement
comparing the three triangles.
Sketch the three right triangles with
the angles of the triangles in
corresponding positions.
By Theorem 8.1, ∆LJK ~ ∆JMK ~ ∆LMJ.
Consider the proportion
. In this case, the
means of the proportion are the same number, and
that number is the geometric mean of the extremes.
The geometric mean of two positive numbers is the
positive square root of their product. So the geometric
mean of a and b is the positive number x such
that
, or x2 = ab.
Example 2A: Finding Geometric Means
Find the geometric mean of each pair of
numbers. If necessary, give the answer in
simplest radical form.
4 and 25
Let x be the geometric mean.
x2 = (4)(25) = 100
x = 10
Def. of geometric mean
Find the positive square root.
Example 2B: Finding Geometric Means
Find the geometric mean of each pair of
numbers. If necessary, give the answer in
simplest radical form.
5 and 30
Let x be the geometric mean.
x2 = (5)(30) = 150
Def. of geometric mean
Find the positive square root.
In Your Notes! Example 2a
Find the geometric mean of each pair of
numbers. If necessary, give the answer in
simplest radical form.
2 and 8
Let x be the geometric mean.
x2 = (2)(8) = 16
x=4
Def. of geometric mean
Find the positive square root.
In Your Notes! Example 2b
Find the geometric mean of each pair of
numbers. If necessary, give the answer in
simplest radical form.
10 and 30
Let x be the geometric mean.
x2 = (10)(30) = 300
Def. of geometric mean
Find the positive square root.
In Your Notes! Example 2c
Find the geometric mean of each pair of
numbers. If necessary, give the answer in
simplest radical form.
8 and 9
Let x be the geometric mean.
x2 = (8)(9) = 72
Def. of geometric mean
Find the positive square root.
You can use Theorem 8.1 to write proportions
comparing the side lengths of the triangles formed
by the altitude to the hypotenuse of a right triangle.
All the relationships in red involve geometric means.
Please turn to page 539
Example 3: Finding Side Lengths in Right Triangles
Find x, y, and z.
62 = (9)(x)
x=4
6 is the geometric mean of
9 and x.
Divide both sides by 9.
y2 = (4)(13) = 52 y is the geometric mean of
4 and 13.
Find the positive square root.
z2 = (9)(13) = 117 z is the geometric mean of
9 and 13.
Find the positive square root.
Helpful Hint
Once you’ve found the unknown side lengths,
you can use the Pythagorean Theorem to check
your answers.
Check It Out! Example 3
Find u, v, and w.
92 = (3)(u)
u = 27
9 is the geometric mean of
u and 3.
Divide both sides by 3.
w2 = (27 + 3)(27) w is the geometric mean of
u + 3 and 27.
Find the positive square root.
v2 = (27 + 3)(3)
v is the geometric mean of
u + 3 and 3.
Find the positive square root.
Example 4: Measurement Application
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 4 Continued
Let x be the height of the tree above eye level.
(7.8)2 = 1.6x
7.8 is the geometric mean of
1.6 and x.
x = 38.025 ≈ 38 Solve for x and round.
The tree is about 38 + 1.6 = 39.6, or 40 m tall.
In Your Notes! Example 4
A surveyor positions himself
so that his line of sight to
the top of a cliff and his line
of sight to the bottom form
a right angle as shown.
What is the height of the
cliff to the nearest foot?
Continued
Let x be the height of cliff above eye level.
(28)2 = 5.5x
x  142.5
28 is the geometric mean of
5.5 and x.
Divide both sides by 5.5.
The cliff is about 142.5 + 5.5, or
148 ft high.