p.1 Similar Right Triangles
p.2 Similar Right Triangles
Example: Finding the height of a roof.
Investigation
1. Cut an index card along one of its diagonals.
Label the vertices A, B, C.
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.
Roof Height: A roof has a cross section that is a right angle. The
diagram shows the approximate dimensions of this cross section.
3. You should now have 3 right triangles. Compare the triangles.
What special property do they share? Explain.
A. Identify the similar triangles. (It may help to redraw them as
separate triangles.)
B. Find the height h of the roof using what you have learned about
similar triangles.
4. What did you discover?
Our next theorem states: 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.
C
∆CBD ~ ∆ABC, ∆ACD ~ ∆ABC, ∆CBD ~ ∆ACD
A
D
B
p.3 Similar Right Triangles
p.4 Similar Right Triangles
C
A
Write a proportion comparing sides CB, AB, and BD in the first and
third triangles.
D
B
In right ∆ABC, altitude CD is drawn to the hypotenuse, forming two
smaller right triangles that are similar to ∆ABC. The triangles have
been redrawn below as 3 separate pictures. Using colored pencils,
shade the sides that are the same the same color.
__________ is the geometric mean between _______ and ________.
B
C
C
B
D
A
Write a proportion comparing sides AC, AB, and AD in the second and
third triangles.
D
A
C
Write a proportion comparing sides AD, CD, and BD in the first two
triangles.
__________ is the geometric mean of ________ and ________.
__________ is the geometric mean between _______ and ________.
What do you think the geometric mean is in your own words?
p.5 Similar Right Triangles
p.6 Similar Right Triangles
Geometric Mean Theorems
Example 2:
2
C
y
5
Solve for y.
A
D
B
In a right triangle, the _______________ from the right angle to the
hypotenuse divides the hypotenuse into two segments. The length of
the altitude is the _______________ _______________ of the
lengths of the two segments.
BD CD

CD AD
In a right triangle, the altitude from the right angle to the hypotenuse
divides the hypotenuse into two segments. The length of each
__________ of the right triangle is the _______________ _________
of the lengths of the hypotenuse and the segment of the hypotenuse
that is _______________ to the leg.
Application:
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 friend’s eye and the
distance from your friend to the track.
AB AC

AC AD
AB CB

CB DB
Example 1:
x
6
3
Solve for x.
Hint: You must find XY in order to write a proportion using what we have learned.