Heights and Distances Class 10 Most Important Notes & MCQs

Trigonometry Heights and Distances Class 10 Complete Notes (English & Hindi)

Introduction (परिचय)
Learning Objectives (सीखने के उद्देश्य)
What are Heights and Distances? (ऊँचाई एवं दूरी क्या है?)
Real-Life Applications (वास्तविक जीवन में उपयोग)
Important Abbreviations (महत्वपूर्ण संक्षिप्त रूप)
Important Terms (महत्वपूर्ण शब्दावली)
Line of Sight (दृष्टि रेखा)
Angle of Elevation (उन्नयन कोण)
Angle of Depression (अवनमन कोण)
Basic Trigonometric Ratios (मूल त्रिकोणमितीय अनुपात)
Standard Trigonometric Values (मानक त्रिकोणमितीय मान)
Important Formula Chart (महत्वपूर्ण सूत्र)
Key Points for Quick Revision (त्वरित पुनरावृत्ति)
Solved Examples (हल किए गए उदाहरण)
Multiple Choice Questions (MCQs)
Assertion and Reason Questions
Case Study Questions
HOTS Questions
NCERT Exercise Questions with Solutions
Previous Year Questions (PYQs)
Common Mistakes to Avoid
Exam Preparation Tips
Frequently Asked Questions (FAQs)
Conclusion (निष्कर्ष)
RELATED TOPIC : Introduction to Trigonometry

Introduction ( Heights and Distances Class 10 )

English ( Heights and Distances Class 10 )

The chapter Some Applications of Trigonometry helps us calculate the height of tall objects and the distance between two places without measuring them directly.

Using trigonometric ratios such as sin, cos, and tan, we can determine the height of buildings, trees, towers, mountains, bridges, and many other objects.

This chapter is one of the most scoring topics in the Class 10 CBSE Board Examination.

हिन्दी

त्रिकोणमिति के अनुप्रयोग अध्याय में हम बिना सीधे मापे किसी वस्तु की ऊँचाई (Height) तथा दूरी (Distance) ज्ञात करना सीखते हैं।

इस अध्याय में मुख्य रूप से sin, cos तथा tan का प्रयोग किया जाता है।

यह अध्याय बोर्ड परीक्षा के लिए अत्यंत महत्वपूर्ण एवं आसान अंक दिलाने वाला अध्याय है।

Learning Objectives ( Heights and Distances Class 10 )

After completing this chapter, students will be able to

Understand Angle of Elevation
Understand Angle of Depression
Calculate unknown heights
Calculate unknown distances
Solve real-life problems
Apply trigonometric ratios effectively

Real-Life Applications ( Heights and Distances Class 10 )

This chapter is used in

Measuring building height
Finding mountain height
Aircraft navigation
Bridge construction
Surveying
Architecture
Engineering
Satellite communication
Defence
Astronomy

हिन्दी

इस अध्याय का प्रयोग

भवन निर्माण
पुल निर्माण
सेना
सर्वेक्षण
इंजीनियरिंग
खगोल विज्ञान
विमानन
पर्वत की ऊँचाई ज्ञात करने में किया जाता है।

Important Abbreviations

Abbreviation	Full Form	Meaning
LOS	Line of Sight	दृष्टि रेखा
AoE	Angle of Elevation	उन्नयन कोण
AoD	Angle of Depression	अवनमन कोण
H	Height	ऊँचाई
D	Distance	दूरी
Tan	Tangent	स्पर्शज्या
Sin	Sine	ज्या
Cos	Cosine	कोज्या

Important Terms ( Heights and Distances Class 10 )

1. Height

English

Height means the vertical distance from the ground to the top of an object.

हिन्दी

भूमि से किसी वस्तु के शीर्ष तक की लंबवत दूरी को ऊँचाई कहते हैं।

2. Distance

English

Distance means the horizontal distance between two points.

हिन्दी

दो बिंदुओं के बीच क्षैतिज दूरी को Distance कहते हैं।

3. Observer

The person who is looking at the object is called the observer.

हिन्दी: जो व्यक्ति वस्तु को देख रहा हो उसे प्रेक्षक कहते हैं।

Line of Sight (LOS)

English

The imaginary line joining the observer’s eye to the object is called the Line of Sight.

हिन्दी

प्रेक्षक की आँख से वस्तु तक जाने वाली काल्पनिक रेखा को दृष्टि रेखा (Line of Sight) कहते हैं।

Example

A boy looks at the top of a tower.

The line joining his eye and the tower top is called Line of Sight.

Angle of Elevation

English

When the observer looks upward towards an object above eye level, the angle formed between the horizontal line and the line of sight is called the Angle of Elevation.

हिन्दी

जब कोई व्यक्ति अपनी आँखों के स्तर से ऊपर स्थित वस्तु को देखता है, तब क्षैतिज रेखा और दृष्टि रेखा के बीच बनने वाला कोण उन्नयन कोण (Angle of Elevation) कहलाता है।

Examples

Looking at a kite
Glance at an airplane
Looking at a tower
Looking at a mountain

Angle of Depression

English

When the observer looks downward towards an object below eye level, the angle between the horizontal line and the line of sight is called the Angle of Depression.

हिन्दी

जब कोई व्यक्ति अपनी आँखों के स्तर से नीचे स्थित वस्तु को देखता है, तब बनने वाला कोण अवनमन कोण (Angle of Depression) कहलाता है।

Examples

Standing on a building and looking down
Looking from a bridge towards a river
Looking from a hill towards a road

Basic Trigonometric Ratios Used

Ratio	Formula
sin θ	Perpendicular / Hypotenuse
cos θ	Base / Hypotenuse
tan θ	Perpendicular / Base
cosec θ	Hypotenuse / Perpendicular
sec θ	Hypotenuse / Base
cot θ	Base / Perpendicular

Most Important Formula ( Heights and Distances Class 10 )

Height Problems

Therefore

Distance Problems

Using Sine

Using Cosine

Standard Trigonometric Values

θ	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	Not Defined

Important Key Points ( Heights and Distances Class 10 )

Height is always vertical.

Distance is always horizontal.

Line of Sight is always the hypotenuse.

Angle of Elevation is measured upward.

Angle of Depression is measured downward.

Most questions are solved using tan θ.

Draw a rough figure before solving any problem.

Write the given values first.

Use standard trigonometric values wherever possible.

Always mention the unit (m, cm, km).

Solved Example 1

A tree is 20 m high. A person stands 20 m away from it. Find the angle of elevation.

Solution

Height = 20 m

Distance = 20 m

Therefore,

Answer: 45°

हिन्दी

पेड़ की ऊँचाई = 20 m

दूरी = 20 m

tanθ = 1

अतः

उत्तर = 45°

Solved Example 2

The angle of elevation of the top of a building is 60°. The observer is 30 m away. Find the height.

Solution

Answer

हिन्दी

दूरी = 30 m

कोण = 60°

tan60 = H/30

H = 30√3 मीटर

Solved Example 3

A tower is 40 m high. Find the distance if the angle of elevation is 45°.

Solution

Therefore,

Solved Example 4

The angle of elevation is 30°. Distance is 50 m. Find the height.

Solution

Solved Example 5

A kite is flying at a height of 25 m. The angle of elevation is 45°. Find the horizontal distance.

Solution

Answer: 25 m

Exam Tips

Draw a neat right-angled triangle before solving.
Identify the height, base (distance), and hypotenuse (line of sight) correctly.
Memorize the standard trigonometric values (0°, 30°, 45°, 60°, 90°).
Use tan θ whenever height and horizontal distance are involved.
Check the final answer with the correct unit (m, cm, km).

Multiple Choice Questions (MCQs)

Trigonometry Heights and Distances Class 10

MCQ 1

The line joining the observer’s eye to the object is called

A) Height

B) Base

C) Line of Sight

D) Hypotenuse

Answer: C

MCQ 2

The angle formed when looking upward at an object is called

A) Angle of Depression

B) Angle of Elevation

C) Right Angle

D) Straight Angle

Answer: B

MCQ 3

The angle formed when looking downward from a higher place is known as

A) Angle of Elevation

B) Angle of Depression

C) Reflex Angle

D) Complementary Angle

Answer: B

MCQ 4

Which trigonometric ratio is most commonly used in height and distance problems?

A) sin θ

B) cos θ

C) tan θ

D) sec θ

Answer: C

MCQ 5

If the angle of elevation is 45°, then tan 45° equals

A) √3

B) 1

C) 1/√3

D) 2

Answer: B

MCQ 6

The height of a tower is 20 m and the distance from the observer is 20 m. The angle of elevation is

A) 30°

B) 45°

C) 60°

D) 90°

Answer: B

MCQ 7

If tan θ = √3, then θ is

A) 30°

B) 45°

C) 60°

D) 90°

Answer: C

MCQ 8 ( Heights and Distances Class 10 )

The horizontal distance between two points is called

A) Height

B) Base

C) Hypotenuse

D) Line of Sight

Answer: B

MCQ 9

The line of sight always represents the

A) Base

B) Perpendicular

C) Hypotenuse

D) Height

Answer: C

MCQ 10

The height of a building is 10√3 m. The observer is 10 m away. The angle of elevation is

A) 30°

B) 45°

C) 60°

D) 90°

Answer: C

MCQ 11

The value of tan 30° is

A) 1

B) √3

C) 1/√3

D) 0

Answer: C

MCQ 12

The value of tan 60° is

A) √3

B) 1

C) 1/√3

D) 2

Answer: A

MCQ 13

If the angle of elevation increases while the distance remains constant, then the height will

A) Increase

B) Decrease

C) Remain Same

D) Become Zero

Answer: A

MCQ 14

Which ratio is equal to Height/Distance?

A) sin θ

B) cos θ

C) tan θ

D) sec θ

Answer: C

MCQ 15

If tan θ = 1, then θ equals

A) 30°

B) 45°

C) 60°

D) 90°

Answer: B

MCQ 16

Which of the following is NOT a trigonometric ratio?

A) tan

B) sin

C) cos

D) area

Answer: D

MCQ 17

An observer looks at the top of a pole. The pole is

A) Horizontal

B) Vertical

C) Slant

D) Curved

Answer: B

MCQ 18

The observer stands 30 m from a building. The angle of elevation is 45°. The height of the building is

A) 20 m

B) 25 m

C) 30 m

D) 45 m

Answer: C

MCQ 19

Which angle is measured from the horizontal line?

A) Central Angle

B) Angle of Elevation

C) Exterior Angle

D) Interior Angle

Answer: B

MCQ 20

The value of tan 0° is

A) 0

B) 1

C) √3

D) Undefined

Answer: A

MCQ 21

The value of tan 45° is

A) 0

B) 1

C) √3

D) 2

Answer: B

MCQ 22

The value of tan 90° is

A) 1

B) 0

C) Undefined

D) √3

Answer: C

MCQ 23

If a person looks down from a tower, the angle formed is

A) Angle of Elevation

B) Angle of Depression

C) Acute Angle

D) Straight Angle

Answer: B

MCQ 24

A tree casts no shadow at noon. The angle of elevation of the Sun is approximately

A) 30°

B) 45°

C) 60°

D) 90°

Answer: D

MCQ 25

The trigonometric ratio used to calculate horizontal distance is generally

A) tan θ

B) sin θ

C) cos θ

D) sec θ

Answer: A

Assertion and Reason Questions ( Heights and Distances Class 10 )

Question 1

Assertion (A): The line of sight is always the hypotenuse of the right triangle.

Reason (R): The line of sight joins the observer’s eye to the object.

A) Both A and R are true, and R is the correct explanation of A.

B) Both A and R are true, but R is not the correct explanation.

C) A is true, but R is false.

D) A is false, but R is true.

Answer: A

Question 2

Assertion (A): Angle of Elevation is measured downward.

Reason (R): It is formed when the observer looks upward.

A) Both A and R are true.

B) A is true, but R is false.

C) A is false, but R is true.

D) Both are false.

Answer: C

Question 3

Assertion (A): tan θ = Height/Base.

Reason (R): Height is perpendicular to the ground.

A) Both are true, and R explains A.

B) Both are true, but R does not explain A.

C) A is false.

D) R is false.

Answer: B

Question 4

Assertion (A): Height is always measured vertically.

Reason (R): Distance is measured horizontally.

A) Both A and R are true.

B) A is true, R is false.

C) A is false, R is true.

D) Both are false.

Answer: A

Question 5

Assertion (A): Angle of Depression is measured below the horizontal line.

Reason (R): The observer looks downward.

A) Both A and R are true, and R is the correct explanation.

B) Both are true, but R is not the explanation.

C) A is false.

D) R is false.

Answer: A

Case Study Questions ( Heights and Distances Class 10 )

Case Study 1: Lighthouse

A lighthouse is 30 m high. A boat is 30 m away from its base.

Q1. The angle of elevation of the top of the lighthouse is

A) 30°

B) 45°

C) 60°

D) 90°

Answer: B

Q2. Which trigonometric ratio is directly used?

A) sin

B) cos

C) tan

D) sec

Answer: C

Q3. The line joining the boat and the top of the lighthouse is

A) Height

B) Base

C) Line of Sight

D) Radius

Answer: C

Case Study 2: Building

A student stands 25 m away from a building. The angle of elevation to the top is 45°.

Q1. Height of the building is

A) 15 m

B) 20 m

C) 25 m

D) 30 m

Answer: C

Q2. The building represents the

A) Base

B) Height

C) Hypotenuse

D) Radius

Answer: B

Q3. Which ratio is used?

A) tan θ

B) sec θ

C) cos θ

D) sin θ

Answer: A

Higher Order Thinking Skills (HOTS) Questions

HOTS Question 1

A boy observes the top of a tower at an angle of elevation of 30°. He walks 20 m towards the tower and now the angle of elevation becomes 60°. Find the height of the tower.

Solution

Let the initial distance from the tower = x m

After moving 20 m towards the tower, the distance becomes (x − 20) m

From the first position,

From the second position,

Substitute the value of h,

Therefore,

Answer: 10√3 m

हिन्दी

मान लें प्रारम्भिक दूरी = x मीटर

पहले स्थान से

दूसरे स्थान से

दोनों समीकरणों को हल करने पर

ऊँचाई = 10√3 मीटर

HOTS Question 2

A tree breaks due to a storm. The broken part makes an angle of 30° with the ground and touches the ground 8 m away from the base. Find the original height of the tree.

Solution

Distance = 8 m

Length of broken part

Original Height

Answer: 8√3 m

HOTS Question 3

From the top of a 40 m building, the angle of depression of a car is 45°.

Find the distance of the car from the building.

Solution

Angle of depression = Angle of elevation = 45°

Answer: 40 m

HOTS Question 4

A balloon is flying vertically above a point on the ground. From another point 60 m away, its angle of elevation is 30°.

Find the height of the balloon.

Solution

Answer: 20√3 m

HOTS Question 5

A ladder 10 m long is placed against a wall making an angle of 60° with the ground.

Find the height reached on the wall.

Solution

Hypotenuse = 10 m

Answer: 5√3 m

NCERT Important Questions with Solutions