NCERT Class 10 Maths — Chapter 9

Exercise 9.1: Heights and Distances

The classic application chapter. Find heights of towers, widths of rivers, and distances using angle of elevation and depression. Draw the diagram first — always.

angle of elevationangle of depressionheight and distance problems

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Extra Practice Questions

These questions cover the same concepts as Exercise 9.1. Try solving them to build confidence before or after the textbook exercise.

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. The height of the tower is:

A.7(1 + √3) m

B.7√3 m

C.7(√3 - 1) m

D.14 m

A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, what is the height of the wall?

A.15√3 m

B.15/√2 m

C.7.5 m

D.15 m

A car is moving towards a tower. It takes 12 minutes for the angle of depression from the top of the tower to the car to change from 30° to 60°. How much more time will the car take to reach the base of the tower from the point where the angle of depression is 60°?

A.4 minutes

B.6 minutes

C.8 minutes

D.10 minutes

A straight tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance from the foot of the tree to the point where the top touches the ground is 8 m. Find the original height of the tree.

A.8√3 m

B.24 m

C.8(1 + √3) m

D.8(1 + 2/√3) m

To find the distance of a boat from the foot of a lighthouse, an observer on top of the lighthouse measures the angle of depression to the boat. If the height of the lighthouse is known, which trigonometric ratio should be used to find the distance of the boat from the foot of the lighthouse?

A.Sine (sin)

B.Cosine (cos)

C.Tangent (tan)

D.Either Sine or Cosine

The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud from the surface of the lake.

A.120 m

B.180 m

C.60√3 m

D.100 m

From the top of a 75m high lighthouse, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

A.75(√3 - 1) m

B.75(√3 + 1) m

C.75√3 m

D.75(2 - √3) m

An observer 1.6 m tall is 20√3 m away from a tower. The angle of elevation from his eye to the top of the tower is 30°. What is the total height of the tower?

A.21.6 m

B.20 m

C.21.6√3 m

D.20√3 m

Which of the following statements is TRUE regarding angles of elevation and depression?

A.The angle of elevation is always greater than the angle of depression for the same object and observer.

B.The angle of depression is formed when the observer looks upwards from a horizontal line.

C.If an observer at point A looks at an object at point B, the angle of elevation from A to B is equal to the angle of depression from B to A.

D.The line of sight is always horizontal for both angle of elevation and depression.

A balloon is flying at a constant height 'h'. From two points on the ground, 'a' meters apart and on the same side of the balloon, the angles of elevation of the balloon are 45° and 60°. Which of the following is the correct expression for the height 'h' of the balloon?

A.a / (√3 - 1)

B.a√3 / (√3 - 1)

C.a(√3 - 1)

D.a / (√3 + 1)

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Common Mistakes to Avoid

✗Not drawing a clear labelled diagram
✗Confusing angle of elevation with angle of depression
✗Using the wrong trig ratio for the triangle
✗Forgetting to add the observer's height when they're standing on something

Frequently Asked Questions

How many questions are in Exercise 9.1?

Exercise 9.1 has 16 questions on heights and distances using trigonometric ratios.

Is this chapter important for boards?

Very important. A 3-4 mark question from heights and distances appears in most CBSE board papers.

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