Right triangle similarity

This presentation will be presented in Unit 5 Lesson 1 on December 9th.

1. Holt Geometry
8-1Similarity in Right Triangles8-1Similarity in Right Triangles
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz

2. Holt Geometry
8-1Similarity in Right Triangles
Warm Up
1. Write a similarity statement
comparing the two triangles.
Simplify.
2. 3.
Solve each equation.
4. 5. 2x
2
= 50
∆ADB ~ ∆EDC
±5

3. Holt Geometry
8-1Similarity in Right Triangles
Use geometric mean to find segment lengths
in right triangles.
Apply similarity relationships in right triangles
to solve problems.
Objectives

4. Holt Geometry
8-1Similarity in Right Triangles
geometric mean
Vocabulary

5. Holt Geometry
8-1Similarity in Right Triangles
In a right triangle, an altitude drawn from the vertex of the
right angle to the hypotenuse forms two right triangles.

6. Holt Geometry
8-1Similarity in Right Triangles

7. Holt Geometry
8-1Similarity in Right Triangles
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.
By Theorem 8-1-1, ∆UVW ~ ∆UWZ ~ ∆WVZ.
Z
W

8. Holt Geometry
8-1Similarity in Right Triangles
Check It Out!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 8-1-1, ∆LJK ~ ∆JMK ~ ∆LMJ.

9. Holt Geometry
8-1Similarity in Right Triangles
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 meanof 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 x
2
= ab.

10. Holt Geometry
8-1Similarity in Right Triangles
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.
x
2
= (4)(25) = 100 Def. of geometric mean
x = 10 Find the positive square root.

11. Holt Geometry
8-1Similarity in Right Triangles
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
x
2
= (5)(30) = 150
Def. of geometric mean
Find the positive square root.

12. Holt Geometry
8-1Similarity in Right Triangles
Check It Out!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.
x
2
= (2)(8) = 16 Def. of geometric mean
x = 4 Find the positive square root.

13. Holt Geometry
8-1Similarity in Right Triangles
Check It Out!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
x
2
= (10)(30) = 300 Def. of geometric mean
Find the positive square root.

14. Holt Geometry
8-1Similarity in Right Triangles
Check It Out!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
x
2
= (8)(9) = 72 Def. of geometric mean
Find the positive square root.

15. Holt Geometry
8-1Similarity 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.

16. Holt Geometry
8-1Similarity in Right Triangles

17. Holt Geometry
8-1Similarity in Right Triangles
Example 3: Finding Side Lengths in Right Triangles
Find x, y, and z.
6
2
= (9)(x) 6 is the geometric mean of 9
and x.
x = 4 Divide both sides by 9.
y
2
= (4)(13) = 52
y is the geometric mean of 4
and 13.
Find the positive square root.
z
2
= (9)(13) = 117 z is the geometric mean of
9 and 13.
Find the positive square root.

18. Holt Geometry
8-1Similarity in Right Triangles
Once you’ve found the unknown side lengths, you can
use the Pythagorean Theorem to check your answers.
Helpful Hint

19. Holt Geometry
8-1Similarity in Right Triangles
Check It Out!Example 3
Find u, v, and w.
w
2
= (27 + 3)(27) w is the geometric mean of
u + 3 and 27.
9
2
= (3)(u) 9 is the geometric mean of
u and 3.
u = 27 Divide both sides by 3.
Find the positive square root.
v
2
= (27 + 3)(3) v is the geometric mean of
u + 3 and 3.
Find the positive square root.

20. Holt Geometry
8-1Similarity 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?

21. Holt Geometry
8-1Similarity in Right Triangles
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.

22. Holt Geometry
8-1Similarity in Right Triangles
Check It Out!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?

23. Holt Geometry
8-1Similarity in Right Triangles
Check It Out!Example 4 Continued
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 28 is the geometric mean of
5.5 and x.
Divide both sides by 5.5.x 142.5

24. Holt Geometry
8-1Similarity 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
12

25. Holt Geometry
8-1Similarity in Right Triangles
Lesson Quiz: Part II
For Items 3–6, use ∆RST.
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
4
TP