Geometry Angles of Elevation and Depression
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Transcript of Geometry Angles of Elevation and Depression
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CONFIDENTIAL 1
Geometry
Angles of Elevation and Depression
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CONFIDENTIAL 2
Warm up
Find each length:
3
5
z
yx
1) X2) Y3) z
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CONFIDENTIAL 3
Angles of Elevation and Depression
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
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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
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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
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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.
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54
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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
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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
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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.
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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.
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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
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CONFIDENTIAL 12
90 ft
T
S x
7˚
write a tangent ratio.
Multiply both sides by x and divide both sides by tan7˚.
Simplify the expression.
By the Alternate Interior Theorem, m/F = 7˚.
tan7 = 90x
x = 90tan7
x 733 ft
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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˚
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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
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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.
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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
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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.
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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.
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CONFIDENTIAL 19
Now some practice problems for you!
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CONFIDENTIAL 20
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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CONFIDENTIAL 21
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
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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
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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
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CONFIDENTIAL 24
Let’s review
Angles of Elevation and Depression
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.
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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
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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.
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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
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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.
Multiply both sides by x and divide both sides by tan41˚.
Simplify the expression.
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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
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CONFIDENTIAL 30
90 ft
T
S x
write a tangent ratio.
Multiply both sides by x and divide both sides by tan7˚.
Simplify the expression.
By the Alternate Interior Theorem, m/F = 7˚.
tan7 = 90x
x = 90tan7
x 733 ft
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CONFIDENTIAL 31
You did a great job today!You did a great job today!