Similar%20Right%20Triangles

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Similar Right Triangles

• Chapter 8

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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.

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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.

?CBD ?ABC, ?ACD ?ABC, ?CBD ?ACD
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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.

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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.

They ALL look the same! Similar same shape,
different size
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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.

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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
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Example Use Geometric Mean to find the Altitude
of the Triangle
6
x

x
3
18 x2
v18 x
v9 v2 x
3 v2 x
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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
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Example Find y (the right leg of the triangle)
using Geometric Mean
5 2
y

y
2
7
y

y
2
14 y2
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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
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Example Find y (the left leg of the triangle)
using Geometric Mean
35
35
y

y
7
7
245 y2
y
7v5 y
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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.

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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.

??XYW ?YZW ?XZY.
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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.