CC Geometry H
Aim #12: How do we use ratios to find missing side lengths of similar triangles?
Do Now: At a certain time of day, a 12 meter flagpole
casts an 8 meter shadow. Write an equation
that would allow you to find the height, h, of
the tree that has a shadow with length, s.
h
12m
8m
s
*What theorem allows us to justify writing this proportion to solve for h?*
Three methods to solve for missing side lengths of similar triangles
Given ΔABC ~ ΔA'B'C', find the missing side lengths.
C
12
4
A
5
C'
6
B
A'
B'
Method 1 (Scale Factor)
Find the scale factor (image/pre-image); use it to compute the other sides.
Since AC = 4 and corresponding side A'C' = 12, scale factor r = ____
A'B' = ___ AB = ___ ___ = ___
BC = B'C' =
=
r
Method 2 (Ratios Between-Figures)
Write ratios relating lengths of corresponding sides.
Solve for A'B' and BC.
A'B' = A'C'
AB
AC
BC = AC
B'C' A'C'
Method 3 (Ratios Within-Figures)
Write ratios relating side lengths within each triangle.
Solve for A'B' and BC.
A'B' = AB
A'C' AC
BC = B'C'
AC
A'C'
1. A large flagpole stands outside an office building.
Mark realizes that when he looks up from the
ground, 60m away from the flagpole, that the top
of the flagpole and the top of the building line up.
The flagpole is 35m tall, and Mark is
170m from the building.
h
35m
60m
170m
a) Prove that the triangle created by the flagpole and the triangle created by the
building are similar.
b) Determine the height of the building using the scale factor. (Method 1)
c) Determine the height of the building using ratios between similar figures.
(Method 2)
d) Determine the height of the building using ratios within similar figures.
(Method 3)
2. Right triangles A, B, C and D are similar.
8
5
5
4
3
A
1.5
B
C
D
a) Write the ratio that compares the height to the hypotenuse of triangle A.
b) Write the ratio that compares the base to the hypotenuse of triangle A.
c) Write the ratio that compares the height to the base of triangle A.
d) Using triangles A and B, answer the following:
(1) Which ratio can be used to determine the height of triangle B?
(2) Which ratio can be used to determine the hypotenuse of triangle B?
(3) Find the unknown lengths of triangle B.
e) Using triangles A and C, answer the following:
(1) Which ratio can be used to determine the base of triangle C?
(2) Which ratio can be used to determine the hypotenuse of triangle C?
(3) Find the unknown lengths of triangle C.
f) Using triangles A and D, answer the following:
(1) Which ratio can be used to determine the height of triangle D?
(2) which ratio can be used to determine the base of triangle D?
(3) find the unknown lengths of triangle D.
3. Dennis needs to fix a leaky roof on his house but does not own a ladder. He
thinks that a 25-foot ladder will be long enough to reach the roof, but he needs to
be sure before he spends the money to buy one. He chooses a point P on the
ground where he can visually align the roof of his car with the edge of the
house roof. Help Dennis determine if a 25-foot ladder will be long
enough for him to reach his roof. If it is long enough,
determine how far, from the base of the
house, he can place the ladder.
4. Catarina's boat has come untied and floated away on the lake. She is standing
atop a cliff that is 35 feet above the water in a lake. If she stands 10 feet from
the edge of the cliff, she can visually align the top of the cliff with the water at
the back of her boat. Her eye level is 5 1/2 feet above the ground. How far out
from the base of the cliff is Catarina's boat (to the nearest tenth)?
Let's Sum it Up!
• Between-figure ratios compare corresponding side lengths between two
similar figures.
• Within-figure ratios compare two side lengths from one figure of the similar
figures. In other words, one ratio compares side lengths from one triangle,
and a second ratio compares side lengths from a second triangle.
Name: ____________________
Date: _____________
CC Geometry H
HW #12
1. ΔDEF ~ ΔABC. All side length measurements are in centimeters. Use betweenfigure ratios and/or within-figure ratios to determine the unknown side lengths.
2 . Given ΔABC ~ ΔXYZ.
a) Write the ratio that compares the
height AC to the hypotenuse of ΔABC.
b) Write the ratio that compares the
base AB to the hypotenuse of ΔABC.
c) Write the ratio that compares the height AC to the base AB of ΔABC.
d) Use within-figure ratios to find the corresponding height of ΔXYZ.
e) Use between-figure ratios to find the hypotenuse of ΔXYZ.
3. The diagram depicts a cross-section of a downhill course, in which two sections
of the course are similar in steepness (≮A ≅ ≮FEG). The vertical drop AB is 625
feet and the vertical drop EF is 500 feet. If the horizontal distance traveled
from F to G is 480 feet, find the horizontal distance traveled from B to C.
OVER
4. The Olympic snowboarding Cross Course
measures 1135 meters in length, with a
vertical drop of 208 m. During his run down the
course, a snowboarder falls at the vertical
location of 100 m. To the nearest meter, how
far down the course did he travel before falling?
5. Brian is photographing the Washington
Monument and wonders how tall the monument is.
Brian places his 5 ft. camera tripod approximately
100 yd. from the base of the monument. Lying on
the ground, he visually aligns the top of his tripod
with the top of the monument and marks his
location on the ground approximately 2 ft. 9 in.
from the center of his tripod. Use Brian's
measurements to calculate the height of the
Washington monument, to the nearest
tenth of a foot.
Mixed Review:
1) Solve for x and y:
2) Solve for x and justify your solution: