Indirect Measurement - CK

Indirect Measurement
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
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Printed: July 30, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITORS
Annamaria Farbizio
www.ck12.org
C ONCEPT
Concept 1. Indirect Measurement
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Indirect Measurement
Here you’ll learn how to apply your knowledge of similar triangles and proportions to model real-life situations and
to find unknown measurements indirectly.
What if you were standing next to a building and wanted to know how tall the building was? How could you use
your own height and the length of the shadows cast by you and the building to determine the building’s height? After
completing this Concept, you’ll be able to solve problems like this one.
Watch This
MEDIA
Click image to the left for more content.
James Sousa:Indirect MeasurmentUsing Similar Triangles
Guidance
An application of similar triangles is to measure lengths indirectly. You can use this method to measure the width
of a river or canyon or the height of a tall object. The idea is that you model a situation with similar triangles and
then use proportions to find the missing measurement indirectly.
Example A
A tree outside Ellie’s building casts a 125 foot shadow. At the same time of day, Ellie casts a 5.5 foot shadow. If
Ellie is 4 feet 10 inches tall, how tall is the tree?
To solve, start by drawing a picture. We see that the tree and Ellie are parallel, so the two triangles are similar.
4 f t, 10 in
5.5 f t
=
x
125 f t
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The measurements need to be in the same units. Change everything into inches and then we can cross multiply.
58 in
66 in
=
x
1500 in
87000 = 66x
x ≈ 1318.18 in or 109.85 f t
Example B
Cameron is 5 ft tall and casts a 12 ft shadow. At the same time of day, a nearby building casts a 78 ft shadow. How
tall is the building?
To solve, set up a proportion that compares height to shadow length for Cameron and the building. Then solve the
equation to find the height of the building. Let x represent the height of the building.
x
5ft
=
12 f t
78 f t
12x = 390
x = 32.5 f t
The building is 32.5 feet tall.
Example C
The Empire State Building is 1250 ft. tall. At 3:00, Pablo stands next to the building and has an 8 ft. shadow. If he
is 6 ft tall, how long is the Empire State Building’s shadow at 3:00?
Similar to Example B, solve by setting up a proportion that compares height to shadow length. Then solve the
equation to find the length of the shadow. Let x represent the length of the shadow.
6ft
1250 f t
=
8ft
x
6x = 10000
x = 1666.67 f t
The shadow is approximately 1666.67 feet long.
Vocabulary
Two triangles are similar if all their corresponding angles are congruent (exactly the same) and their corresponding
sides are proportional (in the same ratio). Solve proportions by cross-multiplying.
Guided Practice
In order to estimate the width of a river, the following technique can be used. Use the diagram.
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Concept 1. Indirect Measurement
Place three markers, O,C, and E on the upper bank of the river. E is on the edge of the river and OC ⊥ CE. Go
across the river and place a marker, N so that it is collinear with C and E. Then, walk along the lower bank of the
river and place marker A, so that CN ⊥ NA. OC = 50 feet, CE = 30 feet, NA = 80 feet.
1. Is 4OCE ∼ 4ANE? How do you know?
2. Is OCkNA? How do you know?
3. What is the width of the river? Find EN.
Answers:
1. Yes. 6 C ∼
= 6 N because they are both right angles. 6 OEC ∼
= 6 AEN because they are vertical angles. This means
4OCE ∼ 4ANE by the AA Similarity Postulate.
∼ 6 EAN. These are alternate interior angles. When
2. Since the two triangles are similar, we must have 6 EOC =
alternate interior angles are congruent then lines are parallel, so OCkNA.
3. Set up a proportion and solve by cross-multiplying.
30 f t
50 f t
=
EN
80 f t
50(EN) = 2400
EN = 48
The river is 48 feet wide.
Practice
The technique from the guided practice section was used to measure the distance across the Grand Canyon. Use the
picture below and OC = 72 f t,CE = 65 f t, and NA = 14, 400 f t for problems 1 - 3.
1. Find EN (the distance across the Grand Canyon).
2. Find OE.
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3. Find EA.
4. Mark is 6 ft tall and casts a 15 ft shadow. A the same time of day, a nearby building casts a 30 ft shadow. How
tall is the building?
5. Karen and Jeff are standing next to each other. Karen casts a 10 ft shadow and Jeff casts an 8 ft shadow. Who is
taller? How do you know?
6. Billy is 5 ft 9 inches tall and Bobby is 6 ft tall. Bobby’s shadow is 13 feet long. How long is Billy’s shadow?
7. Sally and her little brother are walking to school. Sally is 4 ft tall and has a shadow that is 3 ft long. Her little
brother’s shadow is 2 ft long. How tall is her little brother?
8. Ryan is outside playing basketball. He is 5 ft tall and at this time of day is casting a 12 ft shadow. The basketball
hoop is 10 ft tall. How long is the basketball hoop’s shadow?
9. Jack is standing next to a very tall tree and wonders just how tall it is. He knows that he is 6 ft tall and at this
moment his shadow is 8 ft long. He measures the shadow of the tree and finds it is 90 ft. How tall is the tree?
10. Thomas, who is 4 ft 9 inches tall is casting a 6 ft shadow. A nearby building is casting a 42 ft shadow. How tall
is the building?
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