📑 Topics on this page (Jump to Topic):
📌 1. Basics
- Angle of elevation: looking up from the horizontal. Angle of depression: looking down.
- Angle of elevation = corresponding angle of depression (alternate angles).
| Triangle | Side ratio (opp : adj : hyp) |
|---|---|
| 30° | 1 : √3 : 2 |
| 45° | 1 : 1 : √2 |
| 60° | √3 : 1 : 2 |
📘 Formulas:
- h = base × tanθ
- From two points on the same side at 30° and 60°, gap d → h = (√3/2)d
⚠️ Exam Traps:
- Use tan when height and base are involved; sin/cos when the hypotenuse is.
📌 2. Practice MCQs
📝 Practice MCQs (prepared by SF Prep — not PYQs)
Q1. From a point 40 m from the foot of a tower, the angle of elevation of its top is 45°. The height of the tower is: Practice
Correct answer: (B)
Solution: h = 40 × tan45° = 40 m.
Solution: h = 40 × tan45° = 40 m.
Q2. From 15 m away from the foot of a pole, the angle of elevation of its top is 60°. The height of the pole is: Practice
Correct answer: (C)
Solution: h = 15 × tan60° = 15√3 m.
Solution: h = 15 × tan60° = 15√3 m.
Q3. From the top of a 10 m tower, the angle of depression of a point on the ground is 30°. The distance of the point from the foot of the tower is: Practice
Correct answer: (D)
Solution: tan30° = 10/d → d = 10√3 m.
Solution: tan30° = 10/d → d = 10√3 m.
Q4. The angle of elevation of the top of a tower from a point is 30°. On walking 100 m towards the tower it becomes 60°. The height of the tower is: Practice
Correct answer: (A)
Solution: h = (√3/2)d = (√3/2) × 100 = 50√3 m.
Solution: h = (√3/2)d = (√3/2) × 100 = 50√3 m.
Q5. A 30 m ladder leans against a wall making 30° with the ground. How high up the wall does it reach? Practice
Correct answer: (B)
Solution: Height = 30 × sin30° = 15 m.
Solution: Height = 30 × sin30° = 15 m.
Q6. When the sun's elevation is 60°, the length of the shadow of a 6√3 m high pole is: Practice
Correct answer: (C)
Solution: Shadow = h/tan60° = 6√3/√3 = 6 m.
Solution: Shadow = h/tan60° = 6√3/√3 = 6 m.
Q7. From a point 30 m from the foot of a tower, the angle of elevation of its top is 45°. The height of the tower is: Practice
Correct answer: (D)
Solution: h = 30 × tan45° = 30 m.
Solution: h = 30 × tan45° = 30 m.
Q8. From 30 m away from the foot of a pole, the angle of elevation of its top is 60°. The height of the pole is: Practice
Correct answer: (A)
Solution: h = 30 × tan60° = 30√3 m.
Solution: h = 30 × tan60° = 30√3 m.
Q9. The angle of elevation of the top of a tower from a point is 30°. On walking 60 m towards the tower it becomes 60°. The height of the tower is: Practice
Correct answer: (B)
Solution: h = (√3/2)d = (√3/2) × 60 = 30√3 m.
Solution: h = (√3/2)d = (√3/2) × 60 = 30√3 m.
Q10. A 12 m ladder leans against a wall making 60° with the ground. How high up the wall does it reach? Practice
Correct answer: (C)
Solution: Height = 12 × sin60° = 6√3 m.
Solution: Height = 12 × sin60° = 6√3 m.
Q11. From a point 20 m from the foot of a tower, the angle of elevation of its top is 45°. The height of the tower is: Practice
Correct answer: (D)
Solution: h = 20 × tan45° = 20 m.
Solution: h = 20 × tan45° = 20 m.
Q12. From the top of a 25 m tower, the angle of depression of a point on the ground is 30°. The distance of the point from the foot of the tower is: Practice
Correct answer: (A)
Solution: tan30° = 25/d → d = 25√3 m.
Solution: tan30° = 25/d → d = 25√3 m.
Q13. A 24 m ladder leans against a wall making 60° with the ground. How high up the wall does it reach? Practice
Correct answer: (B)
Solution: Height = 24 × sin60° = 12√3 m.
Solution: Height = 24 × sin60° = 12√3 m.
Q14. When the sun's elevation is 60°, the length of the shadow of a 12√3 m high pole is: Practice
Correct answer: (C)
Solution: Shadow = h/tan60° = 12√3/√3 = 12 m.
Solution: Shadow = h/tan60° = 12√3/√3 = 12 m.
Q15. From a point 100 m from the foot of a tower, the angle of elevation of its top is 45°. The height of the tower is: Practice
Correct answer: (D)
Solution: h = 100 × tan45° = 100 m.
Solution: h = 100 × tan45° = 100 m.
Q16. From 50 m away from the foot of a pole, the angle of elevation of its top is 60°. The height of the pole is: Practice
Correct answer: (A)
Solution: h = 50 × tan60° = 50√3 m.
Solution: h = 50 × tan60° = 50√3 m.
Q17. From the top of a 30 m tower, the angle of depression of a point on the ground is 30°. The distance of the point from the foot of the tower is: Practice
Correct answer: (B)
Solution: tan30° = 30/d → d = 30√3 m.
Solution: tan30° = 30/d → d = 30√3 m.
Q18. The angle of elevation of the top of a tower from a point is 30°. On walking 80 m towards the tower it becomes 60°. The height of the tower is: Practice
Correct answer: (C)
Solution: h = (√3/2)d = (√3/2) × 80 = 40√3 m.
Solution: h = (√3/2)d = (√3/2) × 80 = 40√3 m.
Q19. When the sun's elevation is 60°, the length of the shadow of a 9√3 m high pole is: Practice
Correct answer: (D)
Solution: Shadow = h/tan60° = 9√3/√3 = 9 m.
Solution: Shadow = h/tan60° = 9√3/√3 = 9 m.
Q20. From a point 25 m from the foot of a tower, the angle of elevation of its top is 45°. The height of the tower is: Practice
Correct answer: (A)
Solution: h = 25 × tan45° = 25 m.
Solution: h = 25 × tan45° = 25 m.