GEOMETRY 10-3 Similarity in Right Triangles Warm Up Warm Up Lesson Presentation Lesson Presentation...

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Transcript of GEOMETRY 10-3 Similarity in Right Triangles Warm Up Warm Up Lesson Presentation Lesson Presentation...

Slide 1comparing the two triangles.
Use geometric mean to find segment lengths in right triangles.
Apply similarity relationships in right triangles to solve problems.
Objectives
10-3
Similarity in Right Triangles
In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles.
10-3
Write a similarity statement comparing the three triangles.
Sketch the three right triangles with the angles of the triangles in corresponding positions.
UVW ~ UWZ ~ WVZ.
Write a similarity statement comparing the three triangles.
Sketch the three right triangles with the angles of the triangles in corresponding positions.
LJK ~ JMK ~ LMJ.
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
x2 = (4)(25) = 100
10-3
Example 2B: Finding Geometric Means
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
Let x be the geometric mean.
5 and 30
x2 = (5)(30) = 150
10-3
TEACH! Example 2a
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
2 and 8
x2 = (2)(8) = 16
10-3
TEACH! Example 2b
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
Let x be the geometric mean.
10 and 30
x2 = (10)(30) = 300
10-3
TEACH! Example 2c
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
Let x be the geometric mean.
8 and 9
x2 = (8)(9) = 72
10-3
Similarity in Right Triangles
You can use Theorem 8-1-1 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.
10-3
Find x, y, and z.
62 = (9)(x)
x = 4
y2 = (4)(13) = 52
Find the positive square root.
z2 = (9)(13) = 117
Find the positive square root.
10-3
Similarity in Right Triangles
Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers.
Helpful Hint
w2 = (27 + 3)(27) w is the geometric mean of
u + 3 and 27.
u and 3.
Find the positive square root.
v2 = (27 + 3)(3) v is the geometric mean of
u + 3 and 3.
*
Similarity in Right Triangles
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?
10-3
Example 4 Continued
Let x be the height of the tree above eye level.
x = 38.025 ≈ 38
(7.8)2 = 1.6x
The tree is about 38 + 1.6 = 39.6, or 40 m tall.
7.8 is the geometric mean of 1.6 and x.
Solve for x and round.
10-3
TEACH! 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?
10-3
The cliff is about 142.5 + 5.5, or 148 ft high.
Let x be the height of cliff above eye level.
(28)2 = 5.5x
Divide both sides by 5.5.
x 142.5
Similarity in Right Triangles
Lesson Quiz: Part I
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
1. 8 and 18
2. 6 and 15
3. Write a similarity statement comparing the three triangles.
4. If PS = 6 and PT = 9, find PR.
5. If TP = 24 and PR = 6, find RS.
6. Complete the equation (ST)2 = (TP + PR)(?).
RST ~ RPS ~ SPT