Geometry Angles of Elevation and Depression

CONFIDENTIAL 1

Geometry

Angles of Elevation and Depression

CONFIDENTIAL 2

Warm up

Find each length:

3

5

z

yx

1) X2) Y3) z

CONFIDENTIAL 3


An angle of elevation is the angle formed by a horizontal line and a line of sight to a point

above the line. In the diagram./1 is the angle of elevation from the tower T to the plane P.

Next page

T

P2

1

Angle of elevation

Angle of depression

CONFIDENTIAL 4

An angle of depression is the angle formed by a horizontal line and a line of a sight to a

point below the line./2 is the angle of depression from the plane to the tower.

T

P2

1

Angle of elevation

Angle of depression

CONFIDENTIAL 5

Since horizontal lines are parallel, /1 ≅ /2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the

angle of depression from the other point.

T

P2

1

Angle of elevation

Angle of depression

CONFIDENTIAL 6

Classifying Angle of Elevation and DepressionClassify each angle as an angle of elevation or angle of depression.A ) /3/3 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression.B) /4 /4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

6

54

3

CONFIDENTIAL 7

Now you try!1) Use the diagram to classify each angle as an angle

of elevation or angle of depression.

1a. /5 1b. /6

6

54

3

CONFIDENTIAL 8

Finding Distance by Using Angle of Elevation

An air traffic controller at an airport sights a plane at an angle of elevation of 41˚. The pilot reports that plane’s altitude is 4000 ft. What is the horizontal distance between the plane

and the airport? Round to the nearest foot.

Let x be the horizontal distance between the plane and the airport. Let A represent the airport and P represent the plane

4000 ft

P

A 41˚x Next page

CONFIDENTIAL 9

4000 ft

P

A 41˚x

tan41 = 4000x

x = 4000tan41

x 4601 ft

You are given the side opposite /A, and x is the side adjacent to /A. So write a tangent ratio.

Multiply both sides by x and divide both sides by tan41˚.

Simplify the expression.

CONFIDENTIAL 10

Now you try!

2) What if….? Suppose the plane is at an altitude of 3500ft and the angle of elevation from the airport to the plane is

29˚. What is the horizontal distance between the plane and the airport? Round to the nearest foot.

CONFIDENTIAL 11

Finding Distance by Using Angle of Depression

A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7˚. What is the horizontal

distance between the tower and the fire? Round to the nearest foot.

Let T represent the top of the tower and let F represent the fire. Let x be the horizontal distance between the tower and the fire.

S x

90 ft

T

F

7˚

Next page

CONFIDENTIAL 12

90 ft

T

S x

7˚

write a tangent ratio.



By the Alternate Interior Theorem, m/F = 7˚.

tan7 = 90x

x = 90tan7

x 733 ft

CONFIDENTIAL 13

Now you try!

3) Suppose the ranger sees another fire and the angle of depression to the fire is 3˚. What is the horizontal distance

to this fire? Round to the nearest foot.

90 ft

T

S x

3˚

CONFIDENTIAL 14

Aviation Application

A pilot flying at an altitude of 2.7 km sights two control towers directly in front of her. The angle of depression to the base of one tower is 37˚. The angle of depression to the base of the other tower is 58 ˚. What is the distance between the two

towers? Round to the nearest tenth of a kilometer.

Step: 1 Let P represent the plane and let A and B represent the two towers. Let x be the distance between the towers.

2.7 km

Z

37

58

y x58 37

BAC

QP

Next page

CONFIDENTIAL 15

2.7 km

Z

37

58

y x58 37

BAC

QP

Step 2 Find y.By the Alternate Interior Angle Theorem, m/ CAP = 58˚

In APC, tan58 = 2.7y .

So y = 2.7tan58 1.6871 km.

Next page

CONFIDENTIAL 16

2.7 km

Z

37

58

y x58 37

BAC

QP

Step 3 Find Z.

By the Alternate Interior Angle Theorem, m/ CBP = 37˚

In BPC, tan37 = 2.7z .

So z = 2.7tan37 3.5830 km. Next page

CONFIDENTIAL 17

2.7 km

Z

37

58

y x58 37

BAC

QP

Step 4 Find x.x = z – yx = 3.5830 – 1.6871 = 1.9 km.So the two towers are about 1.9 km. apart.

CONFIDENTIAL 18

Now you try!

4) A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78˚, and the angle of depression to the second airport is 19˚. What is the distance between the

airport? Round to the nearest foot.

CONFIDENTIAL 19

Now some practice problems for you!

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Classify each angle as an angle of elevation or angle of depression.

1 ) /12) /23) /34) /4

4

32

1

Assessment

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5) When the angle of elevation to the sun is 37˚, a flagpole casts a shadow that is 24.2 ft long. What is the

height of the flagpole to the nearest foot?

24.7 ft37

CONFIDENTIAL 22

6) The pilot of a traffic helicopter sights an accident at an angle of depression of 18˚. The helicopter’s altitude is

1560 ft. What is the horizontal distance from the helicopter to the accident? Round to the nearest foot.

18

T

P

1560 ft

CONFIDENTIAL 23

7) From the top of a canyon, the angle of depression to the far side of the river is 58˚, and the angle of

depression to the near side of the river is 74˚.The depth of the canyon is 191 m. What is the width of the

river at the bottom of the canyon? Round to the nearest tenth of a meter.

7458

191 m

CONFIDENTIAL 24

Let’s review


An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram./1 is the angle of elevation from the tower T to the plane P.

An angle of depression is the angle formed by a horizontal line and a line of a sight to a point below the line./2 is the angle of depression from the plane to the tower.

CONFIDENTIAL 25

Since horizontal lines are parallel, /1 ≅ /2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the angle of depression from the other point.

T

P2

1

Angle of elevation

Angle of depression

CONFIDENTIAL 26

Classifying Angle of Elevation and DepressionClassify each angle as an angle of elevation or angle of depression.A /3/3 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression.B /4 /4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

6

54

3

CONFIDENTIAL 27

Finding Distance by Using Angle of Elevation

An air traffic controller at an airport sights a plane at an angle of elevation of 41˚. The pilot reports that plane’s altitude is 4000 ft. What is the horizontal distance between the plane and the airport? Round to the nearest foot.

Let x be the horizontal distance between the plane and the airport. Let A represent the airport and P represent the plane

4000 ft

P

A 41˚x Next page

CONFIDENTIAL 28

4000 ft

P

A 41˚x

tan41 = 4000x

x = 4000tan41

x 4601 ft

You are given the side opposite /A, and x is the side adjacent to /A. So write a tangent ratio.



CONFIDENTIAL 29

Finding Distance by Using Angle of Depression

A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7˚. What is the horizontal distance between the tower and the fire? Round to the nearest foot.

Let T represent the top of the tower and let F represent the fire. Let x be the horizontal distance between the tower and the fire.

90 ft

x

7˚

Next page

CONFIDENTIAL 30

90 ft

T

S x

write a tangent ratio.



By the Alternate Interior Theorem, m/F = 7˚.

tan7 = 90x

x = 90tan7

x 733 ft

CONFIDENTIAL 31

You did a great job today!You did a great job today!

Geometry Angles of Elevation and Depression

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