Similarity in Right TrianglesSimilarity in Right Triangles · 8-1 Similarity in Right Triangles Warm Up 1. Write a similarity statement comparing the two triangles. Simplify. 2. 3

Holt McDougal Geometry

8-1 Similarity in Right Triangles 8-1 Similarity in Right Triangles

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz



8-1 Similarity in Right Triangles

Warm Up 1. Write a similarity statement comparing the two triangles.

Simplify. 2. 3. Solve each equation. 4. 5. 2x2 = 50

∆ADB ~ ∆EDC

±5



Use geometric mean to find segment lengths in right triangles.

Apply similarity relationships in right triangles to solve problems.

Objectives



geometric mean

Vocabulary



In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two 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



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.



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 Def. of geometric mean

x = 10 Find the positive square root.



Example 2B: Finding Geometric Means



5 and 30


Find the positive square root.



Check It Out! Example 2a


2 and 8



x = 4 Find the positive square root.



Check It Out! Example 2b



10 and 30





Check It Out! Example 2c



8 and 9





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.





Example 3: Finding Side Lengths in Right Triangles

Find x, y, and z.

62 = (9)(x) 6 is the geometric mean of

9 and x.

x = 4 Divide both sides by 9.

y2 = (4)(13) = 52 y is the geometric mean of

4 and 13.


z2 = (9)(13) = 117 z is the geometric mean of

9 and 13.




Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers.

Helpful Hint




Find u, v, and w.

w2 = (27 + 3)(27) w is the geometric mean of

u + 3 and 27.

92 = (3)(u) 9 is the geometric mean of

u and 3.

u = 27 Divide both sides by 3.


v2 = (27 + 3)(3) v is the geometric mean of

u + 3 and 3.




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.

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.




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?



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



Lesson Quiz: Part I


1. 8 and 18

2. 6 and 15

12



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

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Similarity in Right TrianglesSimilarity in Right Triangles · 8-1 Similarity in Right Triangles Warm Up 1. Write a similarity statement comparing the two triangles. Simplify. 2. 3