8-5 Angles of Elevation and Depression
You used similar triangles to measure distances
indirectly.
• Solve problems involving angles of elevation
and depression.
• Use angles of elevation and depression to
find the distance between two objects.
Vocabulary
Angle of elevation – the angle formed by a horizontal
line and an observer’s line of sight to an object above
the horizontal line.
Angle of depression – the angle formed by a
horizontal line and an observer’s line of sight to an
object below the horizontal line
CIRCUS ACTS At the circus, a person in the audience at
ground level watches the high-wire routine. A 5-foot-6inch tall acrobat is standing on a platform that is 25 feet
off the ground. How far is the audience member from
the base of the platform, if the angle of elevation from
the audience member’s line of sight to the top of the
acrobat is 27°?
Make a drawing.
Since QR is 25 feet and RS is 5
feet 6 inches or 5.5 feet, QS is
30.5 feet. Let x represent PQ.
Multiply both sides by x.
Divide both sides by tan
Answer:
Simplify.
The audience member is about 60 feet from the
base of the platform.
DIVING At a diving competition, a 6-foot-tall diver stands
atop the 32-foot platform. The front edge of the platform
projects 5 feet beyond the ends of the pool. The pool itself is
50 feet in length. A camera is set up at the opposite end of
the pool even with the pool’s edge. If the camera is angled so
that its line of sight extends to the top of the diver’s head,
what is the camera’s angle of elevation to the nearest
degree?
A. 37°
B. 35°
C. 40°
D. 50°
DISTANCE Maria is at the top of a cliff and
sees a seal in the water. If the cliff is 40 feet
above the water and the angle of
depression is 52°, what is the horizontal
distance from the seal to the cliff, to the
nearest foot?
Make a sketch of the situation.
Since
are parallel, mBAC = mACD by
the Alternate Interior Angles Theorem.
Let x represent the horizontal distance from the seal to the cliff, DC.
C = 52°; AD = 40, and DC = x
Multiply each side by x.
Divide each side by tan 52°.
Luisa is in a hot air balloon 30 feet above the ground. She
sees the landing spot at an angle of depression of 34.
What is the horizontal distance between the hot air
balloon and the landing spot to the nearest foot?
A. 19 ft
B. 20 ft
C. 44 ft
D. 58 ft
Two angles of Elevation or Depression
 Angles of elevation or depression to two
different objects can be used to estimate the
distance between those objects.
 Angles from two different positions of
observation to the same object can be used to
estimate the object’s height.
DISTANCE Vernon is on the top deck
of a cruise ship and observes two
dolphins following each other directly
away from the ship in a straight line.
Vernon’s position is 154 meters above
sea level, and the angles of
depression to the two dolphins are
35° and 36°. Find the distance
between the two dolphins to the
nearest meter.
Understand
ΔMLK and ΔMLJ are right triangles.
The distance between the dolphins is
JK or JL – KL. Use the right triangles
to find these two lengths.
Plan
Because
are horizontal
lines, they are parallel. Thus,
and
because they are alternate interior
angles. This means that
Solve
Multiply each side by JL.
Divide each side by tan
Use a calculator.
Multiply each side by KL.
Divide each side by tan
Use a calculator.
Answer:
The distance between the dolphins is
JK – KL. JL – KL ≈ 219.93 – 211.96,
or about 8 meters.
8-5 Assignment
p. 583, 1-6 all
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