CBSE Class 10 Maths Chapter 10: Construction NCERT Solutions

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NCERT Solutions PDF Class 10 PDF

This chapter provides comprehensive NCERT Solutions for Class 10 Mathematics, focusing on the topic of Constructions. It covers essential techniques for dividing a line segment in a given ratio, a fundamental skill in geometry. The solutions explain the step-by-step process for constructing such divisions, ensuring students understand the underlying principles and practical application. These solutions are designed to clarify the methods, helping students build confidence and accuracy in their geometrical constructions. They serve as an excellent resource for exam preparation, offering clear explanations and accurate guidance to master the construction of geometric figures as per NCERT guidelines.

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Quick info

BoardCBSE
ClassClass 10
SubjectMaths (Exemplar)
Session2026
LanguageEnglish
TypeNCERT Solutions
Chapter10. Construction

Chapter summary

Chapter 10 on Constructions for Class 10 Maths NCERT Solutions focuses on the practical aspect of dividing a line segment internally in a given ratio. The exercise includes multiple-choice questions that test the understanding of the minimum number of points required on a ray and the correct points to join for achieving the desired division. The solutions provide a clear, step-by-step approach to these constructions, reinforcing the theoretical concepts with practical execution.

Learning outcomes

  • Understand the procedure for dividing a line segment in a given ratio.
  • Determine the minimum number of points required on a ray for a specific ratio.
  • Identify the correct points to join for constructing the division of a line segment.
  • Apply geometric construction principles to solve problems.
  • Gain proficiency in using geometric tools for accurate constructions.

Topics covered

Paper topics

  • Division of a line segment in a given ratio
  • Construction of a ray making an acute angle
  • Marking points at equal distances on a ray
  • Joining points to form a dividing line segment
  • Understanding ratio in geometric constructions
  • Multiple Choice Questions on Constructions

Important topics

  • Dividing a line segment in ratio m:n
  • Determining the number of points for division
  • Identifying the correct points to join
  • Construction steps for internal division

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Questions and Solutions

Question 1

To divide a line segment AB in the ratio 5:7, first a ray AX is drawn so that \(\angle BAX\) is an acute angle and then at equal distances points are marked on the ray AX such that the minimum number of these points is

(A) 8

(B) 10

(C) 11

(D) 12

Solution:

To divide a line segment AB in a given ratio, say m:n, we follow a specific construction procedure. First, a ray AX is drawn such that \(\angle BAX\) is an acute angle. Then, points are marked on the ray AX at equal distances. The total number of points required on the ray AX is the sum of the terms in the ratio, i.e., m + n.

In this case, the ratio is 5:7. Therefore, m = 5 and n = 7.

The minimum number of points that must be marked on the ray AX is m + n = 5 + 7 = 12.

These 12 points are typically labeled as \(A_1, A_2, ..., A_{12}\).

The correct option is (D) 12.

Question 2

To divide a line segment AB in the ratio 4:7, a ray AX is drawn first such that \(\angle BAX\) is an acute angle and then points \(A_1, A_2, A_3, ...\) are located at equal distances on the ray AX and the point B is joined to

(A) \(A_{12}\)

(B) \(A_{11}\)

(C) \(A_{10}\)

(D) \(A_9\)

Solution:

The problem requires dividing the line segment AB in the ratio 4:7. This means we need to find a point C on AB such that AC:CB = 4:7.

According to the construction procedure:

  1. Draw a ray AX making an acute angle \(\angle BAX\).
  2. Mark points \(A_1, A_2, A_3, ...\) on the ray AX at equal distances. The total number of points to be marked is the sum of the ratio terms, which is 4 + 7 = 11. So, we mark points \(A_1, A_2, ..., A_{11}\).
  3. The point B is then joined to the last marked point on the ray AX. In this case, the last point is \(A_{11}\).
  4. The line segment \(A_{11}B\) intersects the line segment AB at the point C, which divides AB in the ratio 4:7.

Therefore, the point B is joined to \(A_{11}\).

The correct option is (B) \(A_{11}\).

Question 3

To divide a line segment AB in the ratio 5: 6, draw a ray AX such that \(\angle BAX\) is an acute angle, then draw a ray BY parallel to AX and the points \(A_1, A_2, A_3, ...\) and \(B_1, B_2, B_3, ...\) are located at equal distances on ray AX and BY, respectively. Then the points joined are

(A) \(A_5\) and \(B_6\)

(B) \(A_6\) and \(B_5\)

(C) \(A_4\) and \(B_5\)

(D) \(A_5\) and \(B_4\)

Solution:

The problem asks to divide the line segment AB in the ratio 5:6. This means we need to find a point C on AB such that AC:CB = 5:6.

The construction involves the following steps:

  1. Draw a ray AX such that \(\angle BAX\) is an acute angle.
  2. Mark points \(A_1, A_2, A_3, A_4, A_5\) on ray AX at equal distances. Since the first part of the ratio is 5, we mark 5 points.
  3. Draw a ray BY parallel to AX such that \(\angle ABY = \angle BAX\).
  4. Mark points \(B_1, B_2, B_3, B_4, B_5, B_6\) on ray BY at equal distances. Since the second part of the ratio is 6, we mark 6 points.
  5. Join the point \(A_5\) (the 5th point on AX) to the point \(B_6\) (the 6th point on BY).
  6. The line segment \(A_5B_6\) intersects the line segment AB at a point, let's call it C. This point C divides AB in the ratio 5:6.

Therefore, the points joined are \(A_5\) and \(B_6\).

The correct option is (A) \(A_5\) and \(B_6\).

Common mistakes

  • Incorrectly calculating the total number of points needed on the ray.
  • Joining the wrong points to divide the line segment in the specified ratio.
  • Misinterpreting the ratio when determining which points to connect.
  • Errors in drawing parallel lines or acute angles during construction.

Revision tips

  • Practice drawing rays and marking points accurately for each ratio.
  • Ensure you understand why the sum of the ratio parts gives the total number of points.
  • Review the steps for drawing parallel lines, as this is crucial for the construction.
  • Work through each example and MCQ to solidify the construction process.

Practice MCQs

Q1. To divide a line segment AB in the ratio 5:7, a ray AX is drawn such that \(\angle BAX\) is an acute angle. What is the minimum number of points that must be marked on ray AX at equal distances?

Q2. When dividing a line segment AB in the ratio 4:7, after drawing ray AX and marking points \(A_1, A_2,..., A_{11}\) at equal distances, which point is joined to B?

Q3. For dividing a line segment AB in the ratio 5:6, points \(A_1, A_2,..., A_5\) are marked on ray AX and \(B_1, B_2,..., B_6\) are marked on ray BY (where BY is parallel to AX). Which pair of points should be joined to achieve the division?

Frequently asked questions

What is the main concept covered in CBSE Class 10 Maths Chapter 10: Construction?

The main concept is the construction of a line segment divided internally in a given ratio, using geometric principles and tools.

How do these NCERT Solutions help in exam preparation?

These solutions provide clear, step-by-step explanations for each construction problem, helping students understand the methodology and practice effectively for exams.

What is the significance of marking points on the ray AX in these constructions?

Marking points at equal distances on ray AX allows us to accurately measure and divide the line segment into parts corresponding to the given ratio.

How do I determine the correct points to join when dividing a line segment in a ratio like 5:7?

You need to mark a total of 5+7=12 points on the ray AX. Then, you join the 5th point from A on AX to the 7th point from B on the parallel ray BY (or simply join the 5th point on AX to B if constructing directly).

Are these solutions suitable for students who find geometry challenging?

Yes, the solutions break down complex constructions into simple, manageable steps, making them accessible and helpful for students who need extra clarity in geometry.

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