Heights and Distances is an important application-based topic in Madhyamik Mathematics. In this chapter, students use trigonometric ratios to calculate unknown heights and distances from information such as an angle of elevation, angle of depression, shadow length or the length of a ladder.
This Chapter 24 revision material covers MCQ, True or False, Fill in the Blanks and 5-mark descriptive questions. The problems are presented with clear calculations so that students can understand the method rather than simply memorising the answer.
Heights and Distances: Basic Concept
In problems related to heights and distances, a right-angled triangle is usually formed. The known height, horizontal distance and angle are connected through the trigonometric ratios sin, cos and tan.
Most commonly used relation:
Therefore, if the height and horizontal distance are involved, the tangent ratio is generally the most convenient choice.
Important Terms
| Term | Meaning |
|---|---|
| Line of Sight | The straight line joining the observer's eye and the object being observed. |
| Angle of Elevation | The angle made by the line of sight with the horizontal when the observer looks upward. |
| Angle of Depression | The angle made by the line of sight with the horizontal when the observer looks downward. |
| Horizontal Line | A line parallel to the level ground. |
Multiple Choice Questions – 1 Mark
1. If the angle of elevation of the Sun is 60° and a pole is 12 metres high, what will be the length of its shadow?
(a) 2√3 m (b) √3 m (c) 4√3 m (d) 4/√3 m
2. In a right-angled triangle ABC, ∠B = 90°, AC = 60 m and BC = 30√3 m. What is the value of ∠C?
(a) 30° (b) 60° (c) 45° (d) 90°
3. At what angle of elevation of the Sun will the length of a person's shadow be equal to the person's height?
(a) 30° (b) 45° (c) 60° (d) 90°
True or False – 1 Mark
1. If the length of the shadow of a light post is zero, the angle of elevation of the Sun is 90°.
2. As the angle of elevation of the Sun decreases, the length of an object's shadow also decreases.
When the angle of elevation decreases, the shadow becomes longer.
3. A horizontal line is parallel to the ground level.
4. The shadow of a building is 45 metres long. If the angle of elevation of the Sun is 60°, the height of the building is 45√3 metres.
5. If the angle of elevation of the Sun is 45°, the shadow of a 15-metre-long pole will also be 15 metres long.
Fill in the Blanks – 1 Mark
1. When the angle of elevation of the Sun is 45°, the height of a pole and the length of its shadow are ________.
2. If the angle of elevation of the Sun increases from 30° to 60°, the length of a pole's shadow will ________.
3. When a person looks upward at an object, the angle made by the line of sight with the horizontal is called the ________.
4. When the angle of elevation of the Sun is ________ than 45°, the length of the shadow is less than the height of the object.
5. When a person looks downward at an object, the angle made by the line of sight with the horizontal is called the ________.
Important Formulae for Heights and Distances
| Situation | Useful Relation |
|---|---|
| Height and horizontal distance are known | tan θ = Height / Distance |
| Hypotenuse and height are involved | sin θ = Height / Hypotenuse |
| Hypotenuse and horizontal distance are involved | cos θ = Distance / Hypotenuse |
| Object's height and shadow are equal | θ = 45° |
Long Answer Questions – 5 Marks
Question 1: Two Buildings and a Ladder
Two buildings are situated on opposite sides of a road. The foot of a ladder is placed 8 metres from the base of the first building. When the ladder rests against the first building, it makes an angle of 30° with the horizontal. Keeping the foot of the ladder at the same position, it is placed against the second building and makes an angle of 60° with the horizontal.
Find:
- The length of the ladder.
- The distance of the foot of the ladder from the second building.
- The width of the road.
- The height at which the ladder touches the second building.
Step 1: Find the length of the ladder
Step 2: Distance from the second building
Step 3: Width of the road
Therefore, road width = 8 + 8√3/3 metres.
Step 4: Height reached on the second building
Therefore, the ladder touches the second building at a height of 8 metres.
Question 2: Height of a Chimney
From a point on a horizontal line at the same level as the base of a chimney, a person moves 50 metres towards the chimney. The angle of elevation of the top of the chimney changes from 30° to 60°. Find the height of the chimney.
Let the original distance from the chimney be x metres and let the height of the chimney be h metres.
From the first position:
From the second position:
The distance is now x − 50 metres.
Equating the two values:
Answer: The height of the chimney is 25√3 metres.
Question 3: Telegraph Pole and Its Shadow
When the angle of elevation of the Sun changes from 45° to 60°, the length of the shadow of a telegraph pole changes by 4 feet. Find the length of its shadow when the angle of elevation is 30°. Take √3 = 1.732 approximately.
Let the height of the pole be h feet.
Shadow at 45°:
Shadow at 60°:
The difference between the two shadows is 4 feet:
Shadow at 30°:
Using √3 = 1.732, the shadow is approximately 16.39 feet.
Question 4: Height of a Monument
From the roof of an 18-metre-high five-storey building, the angle of elevation of the top of a monument is 45°, while the angle of depression of the base of the monument is 60°. Find the height of the monument. Take √3 = 1.732 approximately.
Let the horizontal distance between the building and the monument be x metres.
Finding the horizontal distance
Finding the height above the roof
Let the portion of the monument above the level of the roof be h metres.
Finding the total height of the monument
Answer: The height of the monument is approximately 28.39 metres.
Key Points to Remember
- Always draw a simple right-angled triangle before starting a Heights and Distances problem.
- Use tan θ = Height / Horizontal Distance when height and distance are involved.
- If the angle of elevation is 45°, the height of an object and its shadow are equal.
- As the angle of elevation increases, the length of the shadow decreases.
- The angle of elevation is used when an object is viewed upward.
- The angle of depression is used when an object is viewed downward.
- For ladder problems, the ladder represents the hypotenuse of the right-angled triangle.
- In multi-step problems, define the unknown height or distance first and then form the trigonometric equations.
Quick Revision Table
| Concept | Important Point |
|---|---|
| Angle of Elevation | Angle formed when the observer looks upward. |
| Angle of Depression | Angle formed when the observer looks downward. |
| Shadow at 45° | Shadow length = Object height. |
| Increasing Elevation | Shadow length decreases. |
| Decreasing Elevation | Shadow length increases. |
| Main Ratio | tan θ = Perpendicular / Base |
How to Solve Heights and Distances Problems
Most problems from this topic become easier when they are divided into small steps. First identify the object whose height or distance has to be found. Then mark the known angle and measurements and form the corresponding right-angled triangle.
- Identify the unknown height or distance.
- Draw or imagine the right-angled triangle.
- Choose the appropriate trigonometric ratio.
- Substitute the known values.
- Solve the equation carefully.
- Write the final answer with the correct unit.
Final Revision
Heights and Distances is mainly about applying trigonometric ratios to real-life situations such as poles, buildings, ladders, chimneys, monuments and shadows. The key is to convert the information in the question into a right-angled triangle and then select the correct ratio.
For Madhyamik Mathematics preparation, revise the meanings of angle of elevation, angle of depression, line of sight and the basic trigonometric ratios. After that, practise different numerical situations so that you can identify the correct method quickly during the examination.