9.1 Similar Right Triangles - PowerPoint PPT Presentation

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9.1 Similar Right Triangles

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9.1 Similar Right Triangles Objectives/Assignment Solve problems involving similar right triangles formed by the altitude drawn to the hypotenuse of a right triangle. – PowerPoint PPT presentation

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Title: 9.1 Similar Right Triangles


1
9.1 Similar Right Triangles
2
Proportions in right triangles
  • In Lesson 8.4, 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.

3
Activity Investigating similar right triangles.
Group of two or three.
  1. Cut an index card along one of its diagonals.
  2. 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.
  3. You should now have three right triangles.
    Compare the triangles. What special property do
    they share? Explain.
  4. Tape your groups triangles to a piece of paper
    and place in homework bin.

4
What did you discover?
  • In the activity, you may have discovered the next
    theorem.

5
Theorem 9.1
  • 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
6
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.

7
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.
8
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.
9
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 9.1, you
    know that ?CBD ?ACD ?ABC.

10
Write this down!
Notice that CD is the longer leg of ?CBD and the
shorter leg of ?ACD. When you write a proportion
comparing the legs lengths of ?CBD and ?ACD, you
can see that CD is the geometric mean of BD and
AD.
Longer leg of ?CBD.
Shorter leg of ?CBD.
BD
CD

CD
AD
Shorter leg of ?ACD
Longer leg of ?ACD.
11
Copy this down!
Sides CB and AC also appear in more than one
triangle. Their side lengths are also geometric
means, as shown by the proportions below
Shorter leg of ?ABC.
Hypotenuse of ?ABC.
AB
CB

CB
DB
Hypotenuse of ?CBD
Shorter leg of ?CBD.
12
Geometric Mean Theorems
  • Theorem 9.2 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 9.3 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.

BD
CD

CD
AD
AB
CB

CB
DB
AB
AC

AC
AD
13
What does that mean?
6
x
5 2
y


x
3
y
2
18 x2
7
y

v18 x
y
2
14 y2
v9 v2 x
v14 y
3 v2 x
14
Ex. 3 Using Indirect Measurement.
  • MONORAIL TRACK. To estimate the height of a
    monorail track, your friend holds a cardboard
    square at eye level. Your friend lines up the
    top edge of the square with the track and the
    bottom edge with the ground. You measure the
    distance from the ground to your friends eye and
    the distance from your friend to the track.

15
In the diagram, XY h 5.75 is the difference
between the track height h and your friends eye
level. Use Theorem 9.2 to write a proportion
involving XY. Then you can solve for h.
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