Playing with Constructions — Class 6 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 6 Mathematics chapter "Playing with Constructions" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 6 Mathematics chapter "Playing with Constructions" — 8 important questions with detailed answers for CBSE…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
NCERT Solutions for Class 6 Mathematics — Playing with Constructions. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Playing with Constructions — All 8 Questions Solved
Q1. What are the tools used in practical geometry and what do they do?
Practical geometry uses specific instruments for accurate constructions
Step 1: Ruler - used to draw straight lines and measure distances
Step 2: Compass - used to draw circles and arcs, and to mark equal distances
Step 3: Set square - used to draw angles like 90°, 45°, 60°, 30°
Step 4: Protractor - used to measure angles and draw angles of any measure
Final Answer: Ruler (for straight lines), Compass (for arcs/circles), Set square (for standard angles), Protractor (for angle measurement)
Q2. Construct a perpendicular to a line at a given point using ruler and compass.
Steps to construct a perpendicular at a point on a line
Step 1: Draw a line AB and mark point P on it
Step 2: Using compass with P as centre, mark equal distances on both sides - points X and Y
Step 3: With compass more than half of XY, draw arcs above and below the line from X and Y
Step 4: Let these arcs intersect at point C and D
Step 5: Draw a line through C and D passing through P
Step 6: CD is perpendicular to AB at P (CD ⊥ AB)
Final Answer: A perpendicular line is constructed at point P
Q3. How do you construct an angle of 60° using ruler and compass?
Steps to construct 60° angle
Step 1: Draw a ray OA
Step 2: Place compass at O with any radius, mark an arc on OA at point B
Step 3: With same radius and B as centre, draw another arc intersecting the first arc at C
Step 4: Join O and C
Step 5: Angle AOC = 60°
Step 6: This works because triangle OBC is equilateral (OB = BC = OC)
Final Answer: Angle of 60° is constructed
Q4. Bisect an angle of 80° using compass and ruler.
Steps to bisect an 80° angle
Step 1: Draw angle AOB of 80°
Step 2: With O as centre, draw an arc that cuts OA at P and OB at Q
Step 3: With P as centre and radius more than half PQ, draw an arc inside the angle
Step 4: With Q as centre and same radius, draw another arc inside the angle
Step 5: Let these arcs intersect at point R
Step 6: Draw ray OR
Step 7: OR bisects angle AOB. Angle AOR = Angle BOR = 40°
Final Answer: The angle is bisected; each part = 40°
Q5. Construct a perpendicular from a point to a line using compass and ruler.
Steps to construct perpendicular from point to line
Step 1: Draw a line AB and point P outside the line
Step 2: With P as centre, draw an arc intersecting AB at two points X and Y
Step 3: With X as centre and radius more than half of XY, draw an arc below the line
Step 4: With Y as centre and same radius, draw another arc intersecting the first arc at Q
Step 5: Draw line PQ
Step 6: PQ is perpendicular to AB
Final Answer: Perpendicular from point P to line AB is constructed
Q6. What is meant by congruent line segments?
Congruent line segments are segments that have equal length
Step 1: Two line segments are congruent if they can be placed one over the other exactly
Step 2: If AB = CD in length, then AB is congruent to CD (written as AB ≅ CD)
Step 3: Using compass, we can compare and mark equal distances
Step 4: This allows us to construct segments equal to a given segment
Final Answer: Congruent line segments have equal length and can coincide completely
Q7. How do you copy a given line segment using compass and ruler?
Steps to copy a line segment
Step 1: Given line segment AB to copy
Step 2: Draw a line and mark point P on it
Step 3: Place compass at A
Step 4: Adjust compass width to reach B (measure length AB)
Step 5: Without changing the width, place compass at P
Step 6: Draw an arc intersecting the line at point Q
Step 7: PQ = AB (PQ is a copy of AB)
Final Answer: Line segment PQ is constructed equal to AB
Q8. Explain how to construct a right angle (90°) using set square.
Steps to construct right angle using set square
Step 1: Place the set square on the paper with its right angle vertex at the desired point
Step 2: The two sides of the set square form the rays for the angle
Step 3: Draw a line along one side of the set square
Step 4: Draw a line along the other side of the set square
Step 5: The angle formed is exactly 90°
Step 6: This is faster and more accurate for drawing right angles
Final Answer: A 90° angle is constructed using set square
Playing with Constructions — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. Which instrument is primarily used to draw circles and arcs in geometry?
- A. Ruler
- B. Protractor
- C. Compass
- D. Set square
Answer: C. Compass — The compass is a geometric tool specifically designed for drawing circles and arcs of various radii.
MCQ 2. What is the primary function of a ruler (or straightedge) in geometric constructions?
- A. To measure angles
- B. To draw curves
- C. To draw straight line segments
- D. To compare masses
Answer: C. To draw straight line segments — A ruler, also known as a straightedge, is used to draw straight lines and measure their lengths.
MCQ 3. When you construct a perpendicular bisector of a line segment, what two conditions does it satisfy?
- A. It divides the segment into three equal parts and forms any angle.
- B. It divides the segment into two unequal parts and is perpendicular to it.
- C. It divides the segment into two equal parts and is perpendicular to it.
- D. It measures the length of the segment and is parallel to it.
Answer: C. It divides the segment into two equal parts and is perpendicular to it. — A perpendicular bisector cuts a line segment into two equal halves (bisects it) and forms a 90-degree angle (is perpendicular) with it.
MCQ 4. To bisect a given angle using a compass and ruler, what is the very first step after placing the compass point at the vertex of the angle?
- A. Draw an arc outside the angle.
- B. Draw an arc intersecting both arms of the angle.
- C. Measure the angle with a protractor.
- D. Draw a straight line from the vertex.
Answer: B. Draw an arc intersecting both arms of the angle. — The first step in bisecting an angle is to place the compass point at the vertex and draw an arc that intersects both arms of the angle.
MCQ 5. Which of these angles can be constructed using only a compass and ruler without a protractor, directly from a ray?
- A. 70 degrees
- B. 60 degrees
- C. 50 degrees
- D. 80 degrees
Answer: B. 60 degrees — A 60-degree angle is one of the fundamental angles that can be constructed accurately using only a compass and a ruler.
MCQ 6. To construct a perpendicular to a line at a specific point 'P' on that line, what is the initial step using a compass?
- A. Draw a very large arc from point P.
- B. Draw two arcs of the same radius on either side of point P, intersecting the line.
- C. Draw a straight line through point P.
- D. Measure 90 degrees with a protractor at point P.
Answer: B. Draw two arcs of the same radius on either side of point P, intersecting the line. — The first step is to draw two arcs of the same radius, centered at P, on either side of P along the line. This establishes two points equidistant from P.
MCQ 7. If you construct two adjacent 60-degree angles on a straight line, starting from a common vertex, what total angle would you have constructed?
- A. 90 degrees
- B. 120 degrees
- C. 180 degrees
- D. 30 degrees
Answer: B. 120 degrees — Constructing two adjacent 60-degree angles means combining their measures. Therefore, 60 degrees + 60 degrees equals a total of 120 degrees.
MCQ 8. Which two tools are absolutely essential for drawing a straight line segment of a specific measured length?
- A. Compass and Protractor
- B. Ruler and Compass
- C. Ruler and Pencil
- D. Set squares and Protractor
Answer: C. Ruler and Pencil — A ruler is used to measure the length and ensure straightness, while a pencil is used to make the actual mark of the line segment.
MCQ 9. In a compass, which specific part is designed to hold the drawing instrument, such as a pencil lead or a small pencil?
- A. The pivot joint
- B. The pointer end
- C. The adjustment screw
- D. The lead holder
Answer: D. The lead holder — The lead holder is the part of the compass specifically designed to grip and secure the pencil lead or a small pencil for drawing purposes.
MCQ 10. To construct a 90-degree angle at a point P on a line, after drawing arcs of the same radius on either side of P (intersecting the line at A and B), what is the next crucial step?
- A. Draw an arc from A and another from B, with a radius greater than half AB, to intersect above P.
- B. Connect point A to point B with a straight line.
- C. Measure the distance between A and B with a ruler.
- D. Draw a circle centered at point P with any radius.
Answer: A. Draw an arc from A and another from B, with a radius greater than half AB, to intersect above P. — After marking points A and B equidistant from P, the next step is to draw arcs from A and B (with a radius greater than half AB) to find their intersection point, which will define the perpendicular.
MCQ 11. A fundamental property of an angle bisector is that every point on it is equidistant from which of the following?
- A. The vertex of the angle
- B. Any point on the angle
- C. The two arms (sides) of the angle
- D. The endpoint of the ray
Answer: C. The two arms (sides) of the angle — Any point that lies on the angle bisector is at an equal perpendicular distance from both arms (sides) of the angle.
MCQ 12. Every point on the perpendicular bisector of a line segment has a special property regarding its distance from what?
- A. The midpoint of the segment
- B. Any point on the segment
- C. The two endpoints of the segment
- D. The line itself
Answer: C. The two endpoints of the segment — A crucial property of a perpendicular bisector is that any point lying on it is equidistant from the two endpoints of the line segment it bisects.
MCQ 13. To construct a 45-degree angle using only a compass and ruler, what is the final step after successfully constructing a 90-degree angle?
- A. Construct a 30-degree angle adjacent to it.
- B. Extend one of the arms of the 90-degree angle.
- C. Bisect the 90-degree angle.
- D. Construct another 45-degree angle next to it.
Answer: C. Bisect the 90-degree angle. — A 45-degree angle is exactly half of a 90-degree angle. Therefore, bisecting a constructed 90-degree angle yields a 45-degree angle.
MCQ 14. What is the minimum number of distinct arcs required to construct the bisector of a given angle using a compass, once the vertex is established?
- A. 1
- B. 2
- C. 3
- D. 4
Answer: C. 3 — You need one arc from the vertex to cut the arms, and then two more arcs (one from each intersection point on the arms) to find the point that defines the bisector.
MCQ 15. Which construction tool is most suitable for drawing or measuring angles of any arbitrary size, such as 37 degrees or 105 degrees?
- A. Ruler
- B. Compass
- C. Protractor
- D. Set square
Answer: C. Protractor — A protractor is specifically designed and used for measuring angles of any arbitrary degree and for drawing angles of specific measures when constructions are not feasible.
MCQ 16. What is the primary function of a compass in geometric constructions?
- A. To measure angles
- B. To draw straight lines
- C. To draw circles and arcs
- D. To measure lengths
Answer: C. To draw circles and arcs — A compass is specifically designed to draw circles and arcs of a given radius around a central point.
MCQ 17. Which geometric tool is used to draw line segments and measure their lengths?
- A. Protractor
- B. Compass
- C. Ruler (Straightedge)
- D. Set-square
Answer: C. Ruler (Straightedge) — A ruler, also known as a straightedge, is used for drawing straight lines and accurately measuring distances.
MCQ 18. How many endpoints does a line segment have?
- A. One
- B. Two
- C. Three
- D. Zero
Answer: B. Two — A line segment is a part of a line that is bounded by two distinct points, called endpoints.
MCQ 19. If you want to draw a circle with a radius of 5 cm, what distance should you set on your compass?
- A. 2.5 cm
- B. 5 cm
- C. 10 cm
- D. Any arbitrary distance
Answer: B. 5 cm — The compass opening directly corresponds to the radius of the circle you wish to draw. So, for a 5 cm radius, the compass should be set to 5 cm.
MCQ 20. What does the term 'bisect' mean in geometry?
- A. To multiply by two
- B. To divide into two equal parts
- C. To draw a perpendicular line
- D. To draw a parallel line
Answer: B. To divide into two equal parts — To bisect something in geometry means to divide it into two equal or congruent parts, such as bisecting a line segment or an angle.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.