NCERT Solutions for Class 6 Maths Chapter 8: Playing with Constructions — Free PDF

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Chapter 8 Overview: Playing with Constructions

Looking for a class 6 math worksheet or maths worksheet for class 6 with answers? This comprehensive resource covers everything you need. Chapter 8 of the NCERT Class 6 Maths textbook (2024-25) introduces students to the art and science of geometric constructions. Using just a ruler (straightedge) and compass, students learn to draw precise geometric figures.

The key topics covered are:
- Constructing a circle — using a compass with a given radius
- Constructing a line segment of a given length
- Copying a line segment using a compass
- Constructing angles — using a protractor and compass-based methods
- Perpendicular bisector of a line segment
- Angle bisector — dividing an angle into two equal parts
- Creating geometric designs and patterns using compass and ruler

Constructions develop precision, spatial reasoning, and an appreciation for the beauty of geometry.

Beyond this worksheet, SparkEd offers interactive online practice for Playing With Constructions with step-by-step solutions and progress tracking — perfect for daily revision.

Exercise 8.1 — Constructing Circles and Line Segments

This exercise covers the basics of using a compass and ruler for constructions.

Problem: Drawing a circle with a given radius

Question: Draw a circle with radius $3.5$ cm.

Solution:

Step 1: Open the compass to $3.5$ cm by placing the metal tip at the $0$ mark of the ruler and adjusting the pencil tip to $3.5$ cm.

Step 2: Mark a point $O$ on the paper. This will be the centre.

Step 3: Place the metal tip of the compass on $O$.

Step 4: Rotate the compass $360°$ to draw the complete circle.

Answer: The circle has centre $O$ and radius $3.5$ cm. Every point on the circle is exactly $3.5$ cm from $O$.

Key terms:
- Radius — distance from centre to any point on the circle ($r = 3.5$ cm)
- Diameter — distance across the circle through the centre ($d = 2r = 7$ cm)
- Circumference — the perimeter of the circle

Problem: Constructing a line segment of given length

Question: Construct a line segment $\overline{AB}$ of length $6.4$ cm.

Solution:

Step 1: Draw a ray starting from point $A$.

Step 2: Place the ruler with the $0$ mark at $A$.

Step 3: Mark point $B$ at the $6.4$ cm mark on the ruler.

Step 4: Draw the segment $\overline{AB}$.

Answer: $\overline{AB} = 6.4$ cm.

Using a compass (alternative):
Step 1: Open the compass to $6.4$ cm using the ruler.
Step 2: Draw a ray from $A$.
Step 3: With the compass on $A$, draw an arc cutting the ray at $B$.
This method is more accurate for longer segments.

Problem: Copying a line segment

Question: Given a line segment $\overline{PQ}$, construct another line segment $\overline{RS}$ of the same length without using a ruler.

Solution:

Step 1: Place the compass tip on $P$ and pencil on $Q$, capturing the length $PQ$.

Step 2: Without changing the compass width, draw a ray from point $R$.

Step 3: Place the compass tip on $R$ and draw an arc cutting the ray at $S$.

Answer: $\overline{RS} = \overline{PQ}$.

This technique is essential because it copies a length exactly without needing to read a measurement.

Exercise 8.2 — Constructing Perpendicular Bisector and Angle Bisector

This exercise introduces two of the most important compass constructions.

Problem: Perpendicular bisector of a line segment

Question: Construct the perpendicular bisector of a line segment $\overline{AB}$ of length $8$ cm.

Solution:

Step 1: Draw $\overline{AB} = 8$ cm.

Step 2: With $A$ as centre and radius more than $\frac{8}{2} = 4$ cm (say $5$ cm), draw arcs above and below $\overline{AB}$.

Step 3: With $B$ as centre and the same radius ($5$ cm), draw arcs above and below $\overline{AB}$, intersecting the first arcs at points $P$ and $Q$.

Step 4: Draw the line through $P$ and $Q$.

Answer: The line $PQ$ is the perpendicular bisector of $\overline{AB}$. It:
- Passes through the midpoint $M$ of $\overline{AB}$ (so $AM = MB = 4$ cm)
- Makes a $90°$ angle with $\overline{AB}$
- Every point on this line is equidistant from $A$ and $B$

Problem: Bisecting a given angle

Question: Construct the bisector of a given angle $\angle AOB = 60°$.

Solution:

Step 1: Draw $\angle AOB = 60°$.

Step 2: With $O$ as centre and any convenient radius, draw an arc cutting $\overrightarrow{OA}$ at $P$ and $\overrightarrow{OB}$ at $Q$.

Step 3: With $P$ as centre and radius $> \frac{PQ}{2}$, draw an arc.

Step 4: With $Q$ as centre and the same radius, draw an arc intersecting the previous arc at $R$.

Step 5: Draw ray $\overrightarrow{OR}$.

Answer: $\overrightarrow{OR}$ bisects $\angle AOB$, so $\angle AOR = \angle ROB = 30°$.

Note: This method works for any angle, not just $60°$.

Problem: Constructing a 60-degree angle

Question: Construct an angle of $60°$ using only a ruler and compass.

Solution:

Step 1: Draw a ray $\overrightarrow{OA}$.

Step 2: With $O$ as centre and any radius $r$, draw an arc cutting $\overrightarrow{OA}$ at $P$.

Step 3: With $P$ as centre and the same radius $r$, draw an arc cutting the first arc at $Q$.

Step 4: Draw ray $\overrightarrow{OQ}$.

Answer: $\angle AOQ = 60°$.

Why it works: Triangle $OPQ$ is equilateral (all sides $= r$), so each angle is $60°$.

From this, you can construct:
- $30° = \frac{60°}{2}$ (bisect $60°$)
- $120° = 2 \times 60°$ (repeat the arc)
- $90°$ (bisect $120°$ and $60°$, or construct a perpendicular)

Exercise 8.3 — Geometric Designs and Patterns

This exercise encourages creativity with geometric constructions.

Problem: Flower pattern with circles

Question: Create a flower pattern by drawing $6$ circles of the same radius, each passing through the centre of the original circle.

Solution:

Step 1: Draw a circle with centre $O$ and radius $r$.

Step 2: Mark a point $A$ on the circle. With $A$ as centre and radius $r$, draw a circle. It passes through $O$ and cuts the original circle at two points.

Step 3: One of these intersection points becomes the centre for the next circle. Repeat this process, moving around the original circle.

Step 4: After $6$ circles, you return to the starting point, forming a petal pattern.

Answer: The resulting figure has $6$ overlapping regions that look like flower petals. This works because $6$ circles of radius $r$ fit exactly around a circle of radius $r$ (since the central angle for each is $60°$).

This is one of the most beautiful constructions in geometry and has appeared in art and architecture for centuries.

Problem: Constructing a regular hexagon

Question: Construct a regular hexagon inscribed in a circle of radius $4$ cm.

Solution:

Step 1: Draw a circle with centre $O$ and radius $4$ cm.

Step 2: Mark any point $A$ on the circle.

Step 3: With compass set to $4$ cm (the radius), place the tip at $A$ and mark point $B$ on the circle.

Step 4: Move to $B$ and mark point $C$. Continue for $D$, $E$, $F$.

Step 5: Connect $A$-$B$-$C$-$D$-$E$-$F$-$A$.

Answer: The hexagon has $6$ equal sides, each $= 4$ cm (equal to the radius).

Why it works: The side of a regular hexagon inscribed in a circle equals the radius of the circle. Each central angle is $\frac{360°}{6} = 60°$, and the triangle formed is equilateral.

Key Concepts and Formulas

Tips for Construction Problems

1. Keep your compass tight. A loose compass changes its radius while drawing, ruining the construction.

2. Use a sharp pencil in the compass for accurate arcs.

3. Do not erase construction arcs. In exams, construction marks carry marks. Show all arcs clearly.

4. Label all points as you construct. This makes it easy to describe your steps.

5. For the perpendicular bisector, the radius of the arcs must be more than half the line segment length. Otherwise, the arcs will not intersect.

6. Practice the flower pattern. It reinforces the compass skill and demonstrates how $60°$ angles connect to circles.

Practice on SparkEd

Constructions develop precision and geometric intuition. SparkEd has 60 practice questions on Playing with Constructions for Class 6 CBSE, with step-by-step solutions and construction guides.

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