Embed Size (px)
Citation preview
CONFIDENTIAL 1
Geometry
Angles of Elevation and Depression
CONFIDENTIAL 2
Warm up
Find each length:
3
5
z
y
x
1) X
2) Y
3) z
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
P
2
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
P
2
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
P
2
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
A41˚
x Next page
CONFIDENTIAL 9
4000 ft
P
A41˚
x
tan41 = 4000
x
x = 4000
tan41
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.
Multiply both sides by x and divide both sides by tan7˚.
Simplify the expression.
By the Alternate Interior Theorem, m/F = 7˚.
tan7 = 90
x
x = 90
tan7
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 = 2.7
y.
So y = 2.7
tan58 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 = 2.7
z.
So z = 2.7
tan37 3.5830 km. Next page
CONFIDENTIAL 17
2.7 km
Z
37
58
y x58 37
BAC
QP
Step 4 Find x.
x = z – y
x = 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!
CONFIDENTIAL 20
Classify each angle as an angle of elevation or angle of depression.
1 ) /1
2) /2
3) /3
4) /4
4
32
1
Assessment
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 ft
37
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
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.
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
P
2
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
A41˚
x Next page
CONFIDENTIAL 28
4000 ft
P
A41˚
x
tan41 = 4000
x
x = 4000
tan41
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 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.
Multiply both sides by x and divide both sides by tan7˚.
Simplify the expression.
By the Alternate Interior Theorem, m/F = 7˚.
tan7 = 90
x
x = 90
tan7
x 733 ft
CONFIDENTIAL 31
You did a great job today!You did a great job today!
