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.