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About Constructions — Class 9 CBSE

Construct bisectors of line segments and angles, and triangles given specific conditions. This topic is part of the CBSE Class 9 mathematics syllabus (chapter: Chapter 11). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-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
  • Bisecting Lines and Angles
  • Constructing Specific Angles
  • Constructing Triangles: Basic Cases
  • Constructing Triangles: Advanced Cases and Summary

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

Constructions — solved examples for Class 9 CBSE

Example 1easy

Which of the following statements is true about the bisector of an angle?
  1. A)It is always perpendicular to one of the arms of the angle.
  2. B)Any point on the bisector is equidistant from the two arms of the angle.
  3. C)It divides the angle into two unequal parts.
  4. D)It always passes through the midpoint of any line segment connecting points on the arms.

Step-by-step solution

  1. An angle bisector is a ray that divides an angle into two equal parts.
  2. A key property of an angle bisector is that any point lying on it is equidistant from the two arms of the angle.
  3. This property is often used in proofs and constructions involving angle bisectors.

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

Example 2medium

A point on the angle bisector of an angle is equidistant from the:
  1. A)vertices of the angle
  2. B)arms of the angle
  3. C)midpoints of the arms
  4. D)exterior of the angle

Step-by-step solution

  1. An angle bisector is a ray that divides an angle into two equal parts.
  2. A fundamental property states that any point on the angle bisector is equidistant from the two arms (or sides) of the angle.

Answer: arms of the angle

Example 3hard

A point P is equidistant from two distinct points A and B. Which of the following statements must always be true about point P?
  1. A)P lies on the line segment AB.
  2. B)P lies on the perpendicular bisector of AB.
  3. C)P lies on the line AB.
  4. D)P is the midpoint of AB.

Step-by-step solution

  1. The perpendicular bisector of a line segment is defined as the locus of all points that are equidistant from the two endpoints of the segment.
  2. If point P is equidistant from A and B (PA = PB), then P must lie on the perpendicular bisector of the segment AB.

Answer: P lies on the perpendicular bisector of AB.

Practice questions on Constructions

  1. Q1.easy

    Shreya wants to construct the perpendicular bisector of a line segment XY. She opens her compass, places the needle at X, and draws an arc above and below XY. Then, she places the needle at Y and draws another arc with a *different* radius, intersecting the first arc. She then draws a line through the intersection points. What mistake did Shreya make?
    1. A)She should have used a radius less than half of XY.
    2. B)She should have only drawn arcs on one side of XY.
    3. C)She should have used the same radius for arcs drawn from X and Y.
    4. D)She should have drawn a circle instead of arcs.
    Show answer

    Answer: She should have used the same radius for arcs drawn from X and Y.

    Hint: Think about the condition for the intersection points to be equidistant from both endpoints of the segment.

  2. Q2.easy

    In classical Euclidean geometric constructions, which of the following tools are permitted?
    1. A)An unmarked ruler and a compass.
    2. B)A marked ruler and a compass.
    3. C)An unmarked ruler and a protractor.
    4. D)A protractor and a compass.
    Show answer

    Answer: An unmarked ruler and a compass.

    Hint: Recall the fundamental tools that form the basis of all classical geometric constructions.

  3. Q3.easy

    To construct an angle of 60° at point O on a given ray OA, which of the following steps is *incorrect*?
    1. A)With O as center and any convenient radius, draw an arc cutting OA at P.
    2. B)With P as center and a radius equal to the first one, draw an arc cutting the previously drawn arc at Q.
    3. C)Draw a ray OQ.
    4. D)The angle AOQ formed will be 60°.
    Show answer

    Answer: With P as center and a radius equal to the first one, draw an arc cutting the previously drawn arc at Q.

    Hint: Consider the geometric shape formed when constructing a 60° angle. All sides of this shape are equal.

  4. Q4.medium

    A point on the perpendicular bisector of a line segment is equidistant from the:
    1. A)endpoints of the line segment
    2. B)perpendicular line itself
    3. C)midpoint of the line segment
    4. D)other points on the line segment
    Show answer

    Answer: endpoints of the line segment

    Hint: Consider the definition and properties of a perpendicular bisector.

  5. Q5.medium

    What is the *first* step when constructing an angle of 60° at a point O on a given ray OA?
    1. A)Draw an arc from point A with a suitable radius.
    2. B)Draw a ray from point O through the intersection of two arcs.
    3. C)With O as center and a suitable radius, draw an arc intersecting OA.
    4. D)Bisect an angle of 120°.
    Show answer

    Answer: With O as center and a suitable radius, draw an arc intersecting OA.

    Hint: Think about how you initiate any angle construction using a compass.

  6. Q6.medium

    To construct an angle of 90° using a compass and ruler, you typically bisect the angle formed by the 60° mark and the _______ mark on the initial arc.
    1. A)30°
    2. B)90°
    3. C)120°
    4. D)180°
    Show answer

    Answer: 120°

    Hint: Think about the two standard angles you construct before making the final 90° angle.

  7. Q7.hard

    Consider an angle ∠XYZ. A point P lies in the interior of ∠XYZ such that the perpendicular distances from P to rays YX and YZ are equal. What can be concluded about point P?
    1. A)P lies on the angle bisector of ∠XYZ.
    2. B)P lies on the line segment XZ.
    3. C)P is equidistant from points X and Z.
    4. D)P is the midpoint of the angle bisector.
    Show answer

    Answer: P lies on the angle bisector of ∠XYZ.

    Hint: Think about the definition of an angle bisector in terms of equidistant points from the arms of the angle.

  8. Q8.hard

    To construct an angle of 67.5° using a ruler and compass, which of the following is the most efficient sequence of steps after constructing a 90° angle?
    1. A)Construct 135°, then bisect it.
    2. B)Bisect 90° to get 45°, then bisect the angle formed by the ray for 45° and the ray for 90°.
    3. C)Bisect 90° to get 45°, then bisect the 45° angle (from 0° to 45°) to get 22.5°, and then sum 45° and 22.5°.
    4. D)Construct 60°, then bisect the angle between 60° and 90° rays.
    Show answer

    Answer: Bisect 90° to get 45°, then bisect the angle formed by the ray for 45° and the ray for 90°.

    Hint: 67.5° is exactly halfway between 45° and 90°. Think about how to find the midpoint of an angle segment.

  9. Q9.hard

    A student constructs a triangle ABC where BC is the base, ∠B is a base angle, and AB + AC is given. After drawing BC and constructing ∠B, the student extends BX such that BD = AB + AC. Which of the following is the next crucial step to locate point A?
    1. A)Construct the perpendicular bisector of CD.
    2. B)Construct the angle bisector of ∠BDC.
    3. C)Construct a line parallel to BC through D.
    4. D)Construct the angle bisector of ∠BCD.
    Show answer

    Answer: Construct the perpendicular bisector of CD.

    Hint: Think about the property that makes the construction work. When BD = AB + AC, and A is on BD, consider triangle ACD.

These are 9 of the 60 questions available for Constructions. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.