Welcome to Part 1 of our new 8-part series on Chapter 9, Applications of Trigonometry (Heights and Distances). This post contains the top 25 Multiple Choice Questions (MCQs) to help you master finding heights, distances, and angles using trigonometric ratios.
Recommended Books for Deep Practice
Top 25 MCQs - Heights and Distances
Question 1: Shadow Length
The height of a tower is 10 m. What is the length of its shadow when the sun's altitude is 45°?
1 = 10 / Shadow
Shadow = 10 m.
Question 2: Angle of Elevation
The angle of elevation of the sun, when the length of the shadow of a tree is √3 times the height of the tree, is:
tan θ = Height / Shadow = h / (h√3) = 1/√3.
tan θ = tan 30° => θ = 30°.
Question 3: Ladder Length
A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, then the height of the wall is:
cos 60° = Height / Hypotenuse
1/2 = H / 15
H = 15/2 m.
(Note: Be careful! Angle is with the *wall*, not ground).
Question 4: Kite String
A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. The length of the string is:
sin 60° = P/H = 60/H
√3/2 = 60/H
H = 120/√3 = (120√3)/3 = 40√3 m.
Question 5: Pole and Shadow
If the height of a vertical pole is √3 times the length of its shadow on the ground, then the angle of elevation of the sun at that time is:
tan θ = Height / Shadow = (√3 * Shadow) / Shadow = √3.
tan θ = tan 60° => θ = 60°.
Question 6: Observer Distance
An observer 1.5 m tall is 28.5 m away from a tower 30 m high. The angle of elevation of the top of the tower from his eye is:
Effective height (P) = 30 - 1.5 = 28.5m.
Distance (Base) = 28.5m.
tan θ = P/B = 28.5/28.5 = 1.
tan θ = 1 => θ = 45°.
Question 7: Broken Tree
A tree is broken by the wind. The top struck the ground at an angle of 30° and at a distance of 30 m from the root. The height of the tree is:
Broken part (Hypotenuse): cos 30 = 30/H => √3/2 = 30/H => H = 60/√3 = 20√3.
Standing part (Perpendicular): tan 30 = P/30 => 1/√3 = P/30 => P = 10√3.
Total Height = P + H = 10√3 + 20√3 = 30√3 m.
Question 8: Depression Angle
From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. The height of the tower is:
tan 45 = 20/x => 1 = 20/x => x = 20.
tan 60 = (20+h)/x => √3 = (20+h)/20.
20√3 = 20 + h => h = 20√3 - 20 = 20(√3 - 1) m.
Question 9: Shadow Change
If the altitude of the sun changes from 30° to 60°, then the length of the shadow of a tower:
Question 10: Two Poles
Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, the distance between their tops is:
Base of right triangle formed = 12m.
Height of right triangle = 11 - 6 = 5m.
Hypotenuse (distance between tops) = √(12² + 5²) = √(144+25) = √169 = 13 m.
Question 11: Angle of Depression
The angle of depression of a car parked on the road from the top of a 150 m high tower is 30°. The distance of the car from the tower is:
tan 30° = Height / Distance
1/√3 = 150 / d
d = 150√3 m.
Question 12: Length of String
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. If the angle made by the rope with the ground level is 30°, then the height of the pole is:
sin 30° = P/H
1/2 = P/20
P = 10 m.
Question 13: Bridge Height
From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°. If the bridge is at a height of 3 m from the banks, then the width of the river is:
Width = x + y.
tan 30 = 3/x => x = 3√3.
tan 45 = 3/y => y = 3.
Width = 3√3 + 3 = 3(√3 + 1) m.
Question 14: Shadow Ratio
The ratio of the length of a rod and its shadow is 1:√3. The angle of elevation of the sun is:
θ = 30°.
Question 15: Balloon Elevation
The angle of elevation of a balloon from a point P on the ground is 60°. After some time, the angle of elevation reduces to 30°. If the balloon is flying at a constant height of 3000√3 m, then the distance travelled by the balloon is:
In 1st triangle (60°): tan 60 = h/x => √3 = 3000√3/x => x = 3000.
In 2nd triangle (30°): tan 30 = h/y => 1/√3 = 3000√3/y => y = 3000(3) = 9000.
Distance travelled = y - x = 9000 - 3000 = 6000 m.
Question 16: Tower Height
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. The height of the tower is:
1/√3 = h/30
h = 30/√3 = 10√3 m.
Question 17: Elevation & Depression
The angles of elevation and depression of the top and bottom of a lighthouse from the top of a 60 m high building are 30° and 60° respectively. The height of the lighthouse is:
tan 60 = 60/Base => √3 = 60/B => B = 20√3.
Now top part: tan 30 = h/B => 1/√3 = h/(20√3) => h = 20.
Total Height = Building + h = 60 + 20 = 80 m.
Question 18: Flagstaff
A 1.6 m tall girl stands at a distance of 3.2 m from a lamp post and casts a shadow of 4.8 m on the ground. The height of the lamp post is:
Height(Girl)/Shadow(Girl) = Height(Lamp)/Total Distance(Shadow+Dist)
1.6/4.8 = H/(3.2 + 4.8)
1/3 = H/8
H = 8/3 m.
Question 19: String Length
A kite is attached to a string. The string makes an angle of 30° with the ground. The height of the kite is 45 m. The length of the string is:
1/2 = 45/String
String = 90 m.
Question 20: Sun's Elevation
At some time of the day, the length of the shadow of a tower is equal to its height. Then the sun's altitude at that time is:
tan θ = H/H = 1.
θ = 45°.
Question 21: Boat Speed
A man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how soon after this will the car reach the observation tower?
Distance at 30° = h√3. Distance at 45° = h.
Distance travelled = h√3 - h = h(√3 - 1).
Time taken = 12 min. Speed = Distance/Time = h(√3-1)/12.
Remaining distance = h.
Time remaining = Dist/Speed = h / [h(√3-1)/12] = 12/(√3-1).
Rationalizing: 12(√3+1)/2 = 6(1.732+1) = 6(2.732) = 16.392 min.
16 min + 0.392*60 sec ≈ 16 min 23 sec.
Question 22: Shadow Calculation
A pole 6 m high casts a shadow 2√3 m long on the ground. Then the sun's elevation is:
tan 60° = √3.
Question 23: Two Ships
Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse observed from the ships are 30° and 45°. If the lighthouse is 100 m high, the distance between the two ships is:
tan 45 = 100/x => x = 100.
tan 30 = 100/y => y = 100√3 = 173.
Total distance = x + y = 100 + 173 = 273 m.
Question 24: Length of Shadow
The shadow of a tower is equal to its height at 10:45 am. The sun's altitude is:
Question 25: String Length 2
A kite is flying at a height of 75 m. The string makes an angle of 60° with the ground. The length of the string is:
√3/2 = 75/L
L = 150/√3 = 50√3 m.
