7.7 Solving Right Triangles with Angles of Elevation

and Depression

Objectives Solve problems using angles of elevation

Solve problems using angles of depression

On top of the Daily Planet, Clark Kent observes the globe tower wobbling. Quickly changing into Superman, he realizes that to save the innocent people below he must quickly…

I must stop his getaway car with my laser beam… What is the angle of depression I need to use to melt his tires and thwart his escape?

Just another sun setting on Metropolis…

“Let your friend, Superman, know that the Daily Planet is about to literally roll down Main Street!”

But as Clark Kent steps out of the Daily Planet, he receives a call from none other than, Lex Luther.

… calculate the angle of elevation necessary to fly directly to the globe and stop it from falling.

Now, why would Luther call and tell me… oh, it’s a diversion, there’s Lex robbing the Main Street Bank.

Now, with Lex Luther in the custody of the police, and after a few geometrical exercises, as we said earlier, it’s just another evening in Metropolis!

Using his superior geometry skills, Superman calculates the angle and distance and flies to the roof top just in time to stop the globe from falling.

Angles of Elevation 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 Superman had to calculate 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 Daily Planet.

x

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 isMake a drawing.

Example 1:

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.

Example 1:

Answer: The audience member is about 60 feet from the base of the platform.

DIVING At a diving competition, a 5 ½ 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

Your Turn:

Angles of Depression 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 Superman from the top of the Daily Planet as he uses his laser vision to thwart Lex Luther.

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 a building and you are looking straight ahead, then you must lower (depress) your eyes to see the street below.

x

Relationship between Angles of Elevation and

Depression As seen in the diagram

to the right, 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 ItemThe angle of depression between the ramp and the horizontal is Use trigonometry to find the height of the ramp.Solve the Test ItemMethod 1The ground and the horizontal level with the platform to which the ramp extends are parallel. Therefore,

since they are alternate interior angles.

Example 2:

Answer: The height of the ramp is about 0.314 meters,

Multiply each side by 3.Simplify.

Y

W

Example 2:

Method 2The 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

Example 2:

Answer: The height of the ramp is about 0.314 meters,

Multiply each side by 3.

Simplify.

Example 2:

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.

Your Turn:

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.

Example 3:

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

Example 3:

Multiply each side by JL.

Divide each side by tan

Use a calculator.

Example 3:

Multiply each side by KL.

Use a calculator.

Divide each side by tan

Answer: The distance between the dolphins is , or about 8 meters.

Example 3:

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

Your Turn:

Assignment Geometry:

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