Heights and Distances Class 10 ICSE — Bansal Chapter 22
The Heights and Distances Class 10 ICSE chapter, from the Bansal Mathematics book, applies trigonometry to real objects. The chapter defines the line of sight, the angle of elevation, and the angle of depression. It then uses the tangent ratio to find heights and distances that cannot be measured directly. This page explains section 22.1 and provides the Bansal Chapter 22 PDF for practice.
ICSE Class 10 Mathematics Bansal Chapter 22 Heights and Distances PDF
What is inside Chapter 22
10 pages · 1 sections · 10 worked examples · 50 exercise questions
| Section | Title |
|---|---|
| 22.1 | Angles of elevation and depression |
- EXERCISE 22(A): 15 questions
- EXERCISE 22(B): 15 questions
- EXERCISE 22(C): 20 questions
22.1 Angles of elevation and depression
Section 22.1 opens with a figure and three definitions. The line of sight joins the observer’s eye to the object being viewed. When the object lies above the observer, the angle the line of sight makes with the horizontal is the angle of elevation. In the textbook figure, angle ACB is the angle of elevation.
The same line of sight gives a second angle when the object lies below the observer. The angle it makes with the horizontal is the angle of depression. In the figure, angle DAC is the angle of depression. The text states a key equality: the elevation angle of A seen from C equals the depression angle of C seen from A.
\angle ACB = \angle DAC
Every problem in this chapter builds one right triangle. The tower or cliff is the vertical side, the ground distance is the horizontal side, and the line of sight is the hypotenuse. So the tangent ratio connects them:
\tan \theta = \frac{\text{height}}{\text{horizontal distance}}
So each question reduces to solving for one side of a right triangle.
Worked pattern. From a point 40\text{ m} from the foot of a tower, the angle of elevation of the top is 45^\circ. Let the height be h metres.
\tan 45^\circ = \frac{h}{40} \quad \Rightarrow \quad 1 = \frac{h}{40} \quad \Rightarrow \quad h = 40\text{ m}
The tower is 40\text{ m} tall. Replace \tan 45^\circ with the tangent of any given angle and solve.
The text also solves examples by a second route. Instead of using the given angle directly, it replaces the angle with its complement, 90^\circ - \theta. In one solved example, \tan 29^\circ is replaced by \tan 61^\circ, and the final distance is identical. Both routes agree.
Common mistake: the angle of elevation is measured at the observer’s eye on the ground, not at the top of the tower. If you place the angle at the wrong vertex, the tangent equation inverts and the answer is wrong.
What to practise from this chapter
The chapter has three exercise blocks. Each question builds the same right-triangle setup from section 22.1.
- EXERCISE 22(A) — 15 questions
- EXERCISE 22(B) — 15 questions
- EXERCISE 22(C) — 20 questions
Exercise 22(A) starts with the sun’s altitude and a tree’s shadow, then moves to ladders, kites, a broken tree and poles on either side of a road. Exercise 22(C) adds figures, a chord in a circle, a flagstaff and an aeroplane.
Several questions carry board-exam year tags. The tags [2004], [2006], [2007], [2008], [2010], [2012], [2014] and [2015] appear in the text. These tagged questions fall into three repeated types:
- a moving observer whose angle of elevation changes as he walks
- a shadow that changes with the sun’s altitude
- two objects, such as ships or climbers, seen from a cliff or tower
Exercise 22(B) question 1 asks for a length without using tables. That question needs the exact values of \tan 30^\circ and \tan 45^\circ. Practising the three repeated types covers most of the chapter’s demand.
For the official syllabus and examination details, check the CISCE website. You can also browse the Class 10 ICSE Maths hub or see all Class 10 ICSE books on this site.
Frequently asked questions
What is the difference between the angle of elevation and the angle of depression?
The angle of elevation is measured when the object is above the observer; the angle of depression is measured when the object is below the observer. Both are the angle the line of sight makes with the horizontal. In the textbook figure, angle ACB is the elevation and angle DAC is the depression.
Why is the angle of elevation equal to the angle of depression?
The text states this equality directly: the angle of elevation of A seen from C equals the angle of depression of C seen from A. Both angles sit between the same line of sight and a horizontal line. So the two angles measure the same amount, \angle ACB = \angle DAC.
How do I find the distance between two ships on the same side of a cliff?
Work out each ship’s distance from the foot of the cliff with the tangent ratio, then combine the two distances. Ships on the same side of the cliff give a distance found by subtraction; ships on opposite sides use addition. The chapter solves a worked example with two ships to show both cases.
Which exercises are in Bansal Chapter 22 Heights and Distances?
The chapter has three exercises: EXERCISE 22(A) with 15 questions, EXERCISE 22(B) with 15 questions, and EXERCISE 22(C) with 20 questions. Exercise 22(B) question 1 asks for a length without using tables.
When do I need trigonometric tables in heights and distances questions?
You need tables when the angle is not a standard one. The solved examples use \tan 29^\circ, \tan 25^\circ, \tan 50^\circ and \tan 32^\circ 24', which come from tables. Exercise 22(B) question 1 is the exception: it says “without using tables”, so only standard angles appear there.
Reference: ICSE Class 10 Mathematics textbook material, chapter 22: Heights and Distances — CISCE-aligned.
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