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SparkEd Math offers 60+ practice questions for Constructions across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

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SparkEd Math uses visual step-by-step solutions for every Constructions problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Constructions content aligned with the ICSE syllabus?

Yes, all Constructions questions and lessons on SparkEd Math are fully aligned with the Indian Certificate of Secondary Education (ICSE) syllabus for Class 9. The content follows the official curriculum structure and exam pattern.

Constructions — Class 9 ICSE (2025-26)

About Constructions — Class 9 ICSE

Construct angle bisectors, perpendicular bisectors, and triangles under given conditions. This topic is part of the ICSE Class 9 mathematics syllabus (chapter: Chapter 16). On this page you can practice 51 questions across three difficulty levels — 20 easy, 20 medium, and 11 hard — each with a visual step-by-step solution, plus a timed 28-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Constructions

Introduction to Geometric Constructions: Tools and Basics
Constructing the Perpendicular Bisector
Constructing the Angle Bisector
Constructing Triangles: Side-Side-Side (SSS) Criterion
Advanced Angles and Exam Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Constructions — solved examples for Class 9 ICSE

Example 1easy

Which of the following statements about an angle bisector is TRUE?

A)Any point on the angle bisector is equidistant from the two vertices of the angle.
B)An angle bisector divides the angle into two unequal parts.
C)Any point on the angle bisector is equidistant from the two arms (sides) of the angle.
D)An angle bisector can only be constructed for acute angles.

Step-by-step solution

An angle bisector is a ray that divides an angle into two equal angles.
A key property is that any point on the angle bisector is equidistant from the two arms (sides) of the angle. This is a fundamental theorem.

Answer: Any point on the angle bisector is equidistant from the two arms (sides) of the angle.

Example 2medium

Which of the following angles can be constructed using only a compass and ruler by drawing a single arc from a point on a line to intersect another arc drawn from the vertex with the same radius?

A)30°
B)45°
C)90°
D)60°

Step-by-step solution

To construct a 60° angle: Draw a ray OA. With O as center and any convenient radius, draw an arc intersecting OA at P. With P as center and the same radius, draw an arc intersecting the previous arc at Q. Join OQ. Angle AOQ is 60°.
This method directly constructs an angle of 60° based on the properties of an equilateral triangle.

Answer: 60°

Example 3hard

A line segment AB is given. A point P is to be constructed such that it is equidistant from A and B, and also equidistant from two distinct intersecting lines L1 and L2. How many such distinct points P can exist at most?

Step-by-step solution

The locus of points equidistant from two points A and B is the perpendicular bisector of the line segment AB. This is a single straight line.
The locus of points equidistant from two distinct intersecting lines L1 and L2 is the pair of angle bisectors of the angles formed by L1 and L2. These two angle bisectors are perpendicular to each other.
A straight line (perpendicular bisector of AB) can intersect a pair of intersecting lines (the angle bisectors of L1 and L2) at most at four distinct points (two intersections with each angle bisector).
Therefore, there can be at most 4 such distinct points P.

Answer: 4

Practice questions on Constructions

Q1.easy
Consider a line segment AB. Which statement accurately describes the property of its perpendicular bisector?
1. A)Any point on the perpendicular bisector is equidistant from the line segment's endpoints A and B.
2. B)The perpendicular bisector passes through one of the endpoints, A or B.
3. C)The perpendicular bisector is parallel to the line segment AB.
4. D)Any point on the perpendicular bisector forms an acute angle with the segment AB.
Show answer
Answer: Any point on the perpendicular bisector is equidistant from the line segment's endpoints A and B.
Hint: Remember that the perpendicular bisector not only divides the segment into two equal parts but also has a special relationship with points lying on it.
Q2.easy
Ravi is constructing a 90° angle at point O on a line L. His steps are:
1. With O as center, draw an arc intersecting L at P and Q.
2. With P as center and a radius *less than* OP, draw an arc.
3. With Q as center and the *same radius* as in step 2, draw another arc intersecting the previous arc at R.
4. Join O to R.
Which step contains an error in the standard construction of a 90° angle?
1. A)Step 1
2. B)Step 2
3. C)Step 3
4. D)Step 4
Show answer
Answer: Step 2
Hint: For constructing perpendiculars and many standard angles, what is the usual relationship between the radius used in step 2 (from P) and the initial radius (OP or PQ)?
Q3.easy
In classical geometric constructions, what are the ONLY two instruments permitted?
1. A)Ruler, Protractor
2. B)Compass, Set-square
3. C)Ruler (unmarked straightedge), Compass
4. D)Protractor, Set-square
Show answer
Answer: Ruler (unmarked straightedge), Compass
Hint: Think about the fundamental tools that define Euclidean geometry constructions.
Q4.medium
A point lying on the angle bisector of an angle is equidistant from:
1. A)The vertex of the angle
2. B)Any point on one arm
3. C)The two arms of the angle
4. D)Any point on the angle
Show answer
Answer: The two arms of the angle
Hint: Think about the definition of an angle bisector in terms of distances from the boundary lines.
Q5.medium
The locus of points equidistant from two fixed points A and B is the:
1. A)Line segment AB
2. B)Circle with center A
3. C)Angle bisector of any angle formed by A and B
4. D)Perpendicular bisector of segment AB
Show answer
Answer: Perpendicular bisector of segment AB
Hint: Consider the geometric shape formed by all points that are the same distance from two specific points.
Q6.medium
To construct a 45° angle using a compass and ruler, the first essential step is to construct a 90° angle. What is the next essential step?
1. A)Bisect the 90° angle
2. B)Construct a 60° angle
3. C)Draw an arc of any radius
4. D)Extend one of the arms
Show answer
Answer: Bisect the 90° angle
Hint: To get half of an angle, which standard construction technique should be used?
Q7.hard
A student constructs a triangle PQR. They then draw the bisector of ∠P and the perpendicular bisector of the side QR. What special point of the triangle PQR is formed by the intersection of these two construction lines *if and only if* triangle PQR is isosceles with PQ = PR?
1. A)Centroid
2. B)Incenter
3. C)Orthocenter
4. D)Circumcenter
Show answer
Answer: Circumcenter
Hint: Consider the properties of isosceles triangles and where the angle bisector of the vertex angle and the perpendicular bisector of the base intersect in such a triangle.
Q8.hard
To construct a triangle ABC given base BC = 7 cm, ∠B = 75°, and the sum of the other two sides AB + AC = 13 cm, a crucial initial step involves:
1. A)Drawing a line segment BX such that ∠CBX = 75°, and cutting off BD = 13 cm on BX.
2. B)Drawing a line segment BX such that ∠CBX = 75°, and cutting off BD = 7 cm on BX.
3. C)Drawing a line segment CX such that ∠BCX = 75°, and cutting off CE = 13 cm on CX.
4. D)Constructing the perpendicular bisector of BC first.
Show answer
Answer: Drawing a line segment BX such that ∠CBX = 75°, and cutting off BD = 13 cm on BX.
Hint: To use the sum of two sides (AB + AC), extend one of the sides forming the given angle and mark the total length.
Q9.hard
To construct a triangle PQR where base QR = 6 cm, ∠Q = 60°, and the difference PQ - PR = 2 cm (PQ > PR), which of the following steps is essential after drawing QR and constructing ∠RQX = 60°?
1. A)Mark a point S on QX such that QS = 2 cm, then join RS.
2. B)Mark a point S on the ray QX extended backwards such that QS = 2 cm, then join RS.
3. C)Mark a point S on QX such that QS = 6 cm, then join RS.
4. D)Construct the perpendicular bisector of QR.
Show answer
Answer: Mark a point S on QX such that QS = 2 cm, then join RS.
Hint: When dealing with the difference of two sides, the difference is marked on the arm of the given angle, or its extension.

These are 9 of the 51 questions available for Constructions. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.