Transcript Angles of Elevation

7.5 Angles of Elevation
and Depression
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Solve problems using angles of elevation
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Solve problems using angles of depression
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An angle of elevation
is the angle between
the line of sight and
the horizontal when
an observer looks
upward.
The angle of elevation is always measured from the
ground up. It is always INSIDE the triangle.
In the diagram above, x marks the angle of elevation to the top
of the tree as seen from a point on the ground. You can think of
the angle of elevation in relation to the movement of your
eyes. If you are looking straight ahead, then you must raise
(elevate) your eyes to see the top of the tree.
CIRCUS ACTS At the circus, a person in the audience
watches the high-wire routine. A 5-foot-6-inch 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
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
Simplify.
Answer: 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?
Answer: about
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An angle of depression
is the angle between
the line of sight when
an observer looks
downward and the
horizontal.
The angle of depression is always OUTSIDE the triangle. It
is never inside the triangle.
In the diagram above, x marks the angle of depression of a boat at sea from
the top of a lighthouse.
Again, you can think of the angle of depression in relation to the movement of
your eyes. If you are standing at the top of the lighthouse and you are
looking straight ahead, then you must lower (depress) your eyes to see the
boat in the water.
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As seen in the diagram,
the dark black horizontal
line is parallel to side CA
of triangle ABC. This
forms two alternate
interior angles which are
equal in measure. Thus,
the angle of elevation = the angle of depression
SHORT-RESPONSE TEST ITEM
A wheelchair ramp is 3 meters long and inclines at
Find the height of the ramp to the nearest tenth
centimeter.
Read the Test Item
The angle of depression between the ramp and the
horizontal is
Use trigonometry to find the height of
the ramp.
Solve the Test Item
Method 1
The ground and the horizontal level with the platform
to which the ramp extends are parallel. Therefore,
since they are alternate interior
angles.
Y
W
Mulitply each side by 3.
Simplify.
Answer: The height of the ramp is about 0.314 meters,
Method 2
The horizontal line from the top of the platform to which
the wheelchair ramp extends and the segment from the
ground to the platform are perpendicular. So,
and
are complementary angles. Therefore,
Y
W
Multiply each side by 3.
Simplify.
Answer: The height of the ramp is about 0.314 meters,
SHORT-RESPONSE TEST ITEM
A roller coaster car is at one of its highest points. It
drops at a
angle for 320 feet. How high was the
roller coaster car to the nearest foot before it began
its fall?
Answer: The roller coaster car was about 285 feet above
the ground.
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
Find the distance between the two dolphins to the
nearest meter.
are right triangles. The distance between
the dolphins is JK or
Use the right triangles to find
these two lengths.
Because
are horizontal lines, they are parallel.
Thus,
and
because they
are alternate interior angles. This means that
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
, or about 8 meters.
Madison looks out her second-floor window, which is
15 feet above the ground. She observes two parked
cars. One car is parked along the curb directly in front
of her window, and the other car is parked directly
across the street from the first car. The angles of
depression of Madison’s line of sight to the cars are
Find the distance between the two cars.
Answer: about 24 feet
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Pre-AP Geometry:
Pg. 373 #4 – 16, 20, 26
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Geometry:
Pg. 373 #4 - 16