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Geometry
8.2 Heights and Distances

Introduction
One of the most important applications of trigonometry is the application of measurement of heights and distances which cannot be measured directly.
It is also extensively used by astronomers in determining the distance of the heavenly bodies like the sun and moon and stars.

Two angles are very often used in the practical applications of trigonometry .
Line of sight
The line of object is the line from our eyes to object.

Angle of Elevation
If the object is above the horizontal level of our eyes, we have to turn our head upwards to view the object. This is called the angle of elevation.

Angle of Depression
If the object is below the horizontal level of our eyes, we have to bend our head to view it. This is called the angle of elevation.

Example 1: The angle of elevation of the top of a tower at a distance of 100 metres from its foot on a horizontal plane is found to be 60o. Find the height of the tower.
Let CA be the tower equal to h metres in length and B point at a distance of 100 metres from its foot C. It is given that angle ABC = 60o

From the right angle triangle ABC we have
h/100 = tan 60
Solving the above we get h = 173.2 metres
Answer: Height of the tower is 173.2 metres

Example 2: From the top of a cliff, 200 metres high, the angle of depression of the top and bottom of a tower are observed to be observed to be 30o and 60o, find the height of the tower.

Solution: Let AB represent the tower and P the top of cliff MP. If PX be the horizontal line through P, then angle XPA = 30o and angle XPB = 60o. Let the height of the tower be h metres. From A draw AL perpendicular to PM.
ML = AB = h=> LP = (200-h)
Again, ang PBM = ang XPB = 60
ang PAL = ang XPA = 30 (alternate angles )
From the right triangle PMB, BM/200 = cot 60
BM = 200 cot 60 = 200/3
From the right triangle PLA, AL /AP = cot 30 =>AL = LP cot 30 = =(200-h)3
But AL = BM = =(200-h)3 =200/3
Answer: h = 133 1/3 metres


Directions: Answer the following questions. Also write at least 10 examples of your own.
Q 1: The angular elevation of a tower from a point is 30 degrees at a point in a horizontal line to the foot of the tower and 100 metres near it is 60 degrees, find the distance of the first point from the tower.
100 metres
150 metres
86.6 metres

Q 2: The elevation of a tower at a point 60 metres from it is cot-13/5 Obtain the height of the tower.
45 metres
110 metres
100 metres

Q 3: A person standing on the bank of a river, observes that the angle subtended by a tree on the opposite bank is 60 degrees, when he retires 40 metres from the bank, he finds the angle to be 30 degrees. Find the height of the tree.
64 metres
34 metres
20 metres

Q 4: What is the angle of elevation of the sun when the length of the shadow of a pole is 31/2 times the height of the pole?
30o
60o
90o

Q 5: From the light house the angles of depression of two ships on opposite sides of the light house are observed to be 30o and 45o. If the height of the light house is 300 metres, find the distance between the ships if the line joining them passes through foot of the light house.
800 metres
809 metres
819.6 metres

Q 6: A vertical flagstaff stands on a horizontal plane from a point distant 150 metres from its foot, the angle of elevation of its top is found to be 30o, find the height of the flagstaff.
113.4 metres
100 metres
100.4 metres

Q 7: If the shadow of a tower is 30 metres, when the sun's altitude is 30o, what is the length of the shadow when the sun's altitude is 60o?
200 metres
240 metres
100 metres

Q 8: The angle of elevation of the top of a tower which is yet incomplete at a point 120 metres from its base is 45o. How much higher should it be raised so that the elevation at the same point may become 60o?
87.84 metres
100 metres
36.85 metres

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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