CBSE Class 9 Maths Chapter 11 Constructions NCERT Solutions

Reviewed by · Updated on

NCERT Solutions PDF Class 9 PDF

This chapter, "Constructions," from CBSE Class 9 Maths, delves into the essential geometric techniques of creating angles and triangles using only a ruler and compass. The NCERT Solutions offer detailed, step-by-step instructions for these constructions. Students will learn about which angles can be constructed and the specific conditions needed to construct a triangle when given two sides and an angle, or when the difference between two sides and an angle is provided. These solutions aim to enhance students' understanding of how geometric theorems are applied in practice and improve their skills in performing precise constructions. Mastering these techniques is vital for exam success and building a solid foundation in geometry.

Home

Quick info

BoardCBSE
ClassClass 9
SubjectMaths Exemplar
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 11

Chapter summary

Chapter 11, "Constructions," for Class 9 Maths NCERT Solutions covers the essential techniques for drawing geometric figures using a ruler and compass. It addresses the constructibility of specific angles and the conditions under which a triangle can be formed, particularly when dealing with the difference between two sides and a given angle. The exercises focus on applying these principles to solve problems related to geometric constructions.

Learning outcomes

  • Understand the conditions for constructing specific angles using a ruler and compass.
  • Identify angles that cannot be constructed with basic geometric tools.
  • Apply the triangle inequality theorem to determine the possibility of constructing a triangle.
  • Solve problems related to constructing triangles given specific side lengths and angles.
  • Differentiate between constructible and non-constructible triangles based on given parameters.

Topics covered

Paper topics

  • Geometric Constructions
  • Constructing Angles
  • Constructible Angles
  • Triangle Construction
  • Conditions for Triangle Construction
  • Difference of Sides in Triangle Construction
  • Ruler and Compass Constructions
  • Geometric Theorems Application

Important topics

  • Conditions for constructing a triangle
  • Constructibility of angles
  • Applying triangle inequality for construction
  • Difference of two sides and triangle construction

PDF preview

Read page by page below. PDF is streamed from the official NCERT website — no download button on this page.

Loading document …
Page of
Loading page …

Questions and Solutions

Multiple Choice Questions: 1

With the help of a ruler and a compass, it is not possible to construct an angle of:

(A) 37.5^{\circ}

(B) 40^{\circ}

(C) 22.5^{\circ}

(D) 67.5^{\circ}

Solution:

Using a ruler and compass, we can construct angles that are multiples of 15° (like 90°, 60°, 30°, 45°, 75°, etc.) and angles obtained by bisecting these (like 45°, 22.5°, 7.5°, etc.). Angles like 37.5° (75^{\circ}/2), 22.5° (45^{\circ}/2), and 67.5° (135^{\circ}/2 or 90^{\circ} + 22.5^{\circ}) are constructible. However, an angle of 40° cannot be constructed using only these basic tools. Therefore, the correct option is (B).

Multiple Choice Questions: 2

The construction of a triangle ABC, given that BC = 6 cm and \angle B = 45°, is not possible when the difference of AB and AC is equal to:

(A) 6.9 cm

(B) 5.2 cm

(C) 5.0 cm

(D) 4.0 cm

Solution:

Given: BC = 6 cm and \angle B = 45°.

For the construction of a triangle, a fundamental condition is that the difference between the lengths of any two sides must be less than the length of the third side. Mathematically, for sides AB, AC, and BC, we must have:

|AB - AC| < BC

|AB - BC| < AC

|AC - BC| < AB

In this case, we are given BC = 6 cm. If the difference between AB and AC is equal to or greater than BC (i.e., |AB - AC| \ge BC), the triangle cannot be constructed. Among the given options, 6.9 cm is greater than BC (6 cm). Therefore, if |AB - AC| = 6.9 cm, the construction is not possible. The correct option is (A).

Multiple Choice Questions: 3

The construction of a triangle ABC, given that BC = 3 cm and \angle C = 60°, is possible when the difference of AB and AC is equal to:

(A) 3.2 cm

(B) 3.1 cm

(C) 3 cm

(D) 2.8 cm

Solution:

Given: BC = 3 cm and \angle C = 60°.

For a triangle to be constructible, the difference between the lengths of any two sides must be strictly less than the length of the third side. This means |AB - AC| < BC.

In this problem, BC = 3 cm. We need to find a value for |AB - AC| such that the triangle can be constructed. This requires |AB - AC| < 3 cm.

Let's examine the options:

(A) 3.2 cm: 3.2 > 3, so not possible.

(B) 3.1 cm: 3.1 > 3, so not possible.

(C) 3 cm: 3 = 3, so not possible (difference must be strictly less).

(D) 2.8 cm: 2.8 < 3, so this condition allows for the construction of the triangle.

Therefore, the construction is possible when the difference between AB and AC is 2.8 cm. The correct option is (D).

Common mistakes

  • Assuming all angles are constructible with a ruler and compass.
  • Incorrectly applying the triangle inequality theorem for construction.
  • Confusing the conditions for constructing a triangle when the sum or difference of sides is given.
  • Errors in calculating or comparing side lengths for triangle constructibility.

Revision tips

  • Practice constructing all standard angles (e.g., 90°, 60°, 45°, 30°, 22.5°).
  • Memorize the conditions for triangle construction, especially concerning the difference of two sides.
  • Work through each example and exercise to solidify understanding of the construction steps.
  • Review the reasons why certain constructions are not possible.

Practice MCQs

Q1. Which of the following angles cannot be constructed using only a ruler and a compass?

Q2. For constructing a triangle ABC with BC = 6 cm and \angle B = 45°, the construction is not possible if the difference between sides AB and AC is:

Q3. Construction of a triangle ABC with BC = 3 cm and \angle C = 60° is possible if the difference between sides AB and AC is:

Frequently asked questions

What is the main focus of Chapter 11: Constructions in Class 9 Maths?

Chapter 11 focuses on the practical skills of constructing geometric figures, specifically angles and triangles, using only a ruler and compass, and understanding the conditions that make these constructions possible.

Which angles are generally not possible to construct with a ruler and compass?

Angles that are not multiples of 15° or cannot be derived by bisecting constructible angles are typically not possible to construct using only a ruler and compass.

What is the key condition for constructing a triangle when the difference between two sides is given?

The construction of a triangle is possible only if the difference between the lengths of any two sides is strictly less than the length of the third side.

How do these NCERT Solutions help students?

These solutions provide clear, step-by-step explanations and reasoning for each problem, helping students understand the methods and conditions for geometric constructions and prepare effectively for exams.

Are all angles constructible with a ruler and compass?

No, not all angles are constructible. For example, an angle of 40° cannot be constructed using only a ruler and compass.

Important links

Content reviewed by the NCERT Help team. Editorial Team and update policy

NCERT Solutions PDF PDF on NCERT Help. URL unchanged for search indexing.