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Similar Right Triangles. Chapter 8. Cut an index card along one of its diagonals. On one of the right triangles, draw an altitude from the right angle to the hypotenuse. Cut along the altitude to form two right triangles.

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activity investigating similar right triangles do in pairs or threes
Cut an index card along one of its diagonals.

On one of the right triangles, draw an altitude from the right angle to the hypotenuse. Cut along the altitude to form two right triangles.

You should now have three right triangles. Compare the triangles. What special property do they share? Explain.

Activity: Investigating similar right triangles. Do in pairs or threes
theorem 8 1 page 518
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.Theorem 8.1 (page 518)

∆CBD ~ ∆ABC, ∆ACD ~ ∆ABC, ∆CBD ~ ∆ACD

proportions in right triangles
In chapter 7, you learned that two triangles are similar if two of their corresponding angles are congruent. For example ∆PQR ~ ∆STU. Recall that the corresponding side lengths of similar triangles are in proportion. Proportions in right triangles
using a geometric mean to solve problems
In right ∆ABC, altitude CD is drawn to the hypotenuse, forming two smaller right triangles that are similar to ∆ABC From Theorem 8.1, you know that ∆CBD~∆ACD~∆ABC.

C

C

A

A

D

D

B

B

Using a geometric mean to solve problems

They ALL look the same!

Similar = same shape, different size

geometric mean theorems
Geometric Mean Theorems
  • Theorem 8.2: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments
  • Theorem 8.3: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
geometric mean theorems1
Geometric Mean Theorems

If you want to find the Altitude… use Geometric Mean

BD

CD

=

GM

CD

AD

CD is the Geometric Mean

of AD

and BD

example use geometric mean to find the altitude of the triangle
Example: Use Geometric Mean to find the Altitude of the Triangle

6

x

=

x

3

18 = x2

√18 = x

√9 ∙ √2 = x

3 √2 = x

geometric mean theorems2

A

B

Geometric Mean Theorems

If you want to find the side on the right side of the triangle… use Geometric Mean

GM

AB

CB

=

CB

DB

CB is the Geometric Mean

of DB

and AB

example find y the right leg of the triangle using geometric mean

Y is the leg on the right side

Rotate the triangle to help visualize what part you need

√14 = y

Example: Find y (the right leg of the triangle) using Geometric Mean

5 + 2

y

=

y

2

7

y

=

y

2

14 = y2

geometric mean theorems3

A

B

Geometric Mean Theorems

If you want to find the side on the LEFT side of the triangle… use Geometric Mean

GM

AB

AC

=

AC

AD

AC is the Geometric Mean

of AD

and AB

ex 1 finding the height of a roof
Roof Height. A roof has a cross section that is a right angle. The diagram shows the approximate dimensions of this cross section.

A. Identify the similar triangles.

B. Find the height h of the roof.

Ex. 1: Finding the Height of a Roof
solution
You may find it helpful to sketch the three similar triangles so that the corresponding angles and sides have the same orientation. Mark the congruent angles. Notice that some sides appear in more than one triangle. For instance XY is the hypotenuse in ∆XYW and the shorter leg in ∆XZY.Solution:

∆XYW ~ ∆YZW ~ ∆XZY.

solution for b
Solution for b.
  • Use the fact that ∆XYW ~ ∆XZY to write a proportion.

YW

XY

Corresponding side lengths are in proportion.

=

ZY

XZ

h

3.1

=

Substitute values.

5.5

6.3

6.3h = 5.5(3.1)

Cross Product property

Solve for unknown h.

h≈ 2.7

The height of the roof is about 2.7 meters.