Complete Summary, Explanations, and Solutions for Playing with Constructions – Ganita Prakash Class VI, Chapter 8 – Compass, Ruler, Geometric Constructions, Questions, Answers
Detailed summary and explanation of Chapter 8 'Playing with Constructions' from the Ganita Prakash Mathematics textbook for Class VI, covering artwork using compass and ruler, circles and their properties, constructing squares and rectangles, properties of squares and rectangles, exploring diagonals, points equidistant from two given points, perpendicular lines, breaking rectangles into squares, geometric constructions with step-by-step methods—along with all NCERT questions, answers, and solutions.
This chapter explores geometric constructions using ruler and compass, focusing on artwork, circles, squares, rectangles, diagonals, and explorations in rectangles.
Main Topics Covered
Artwork with freehand and instruments
Circles: Center, radius, drawing
Constructing figures: Person, Wavy Wave, Eyes
Squares and rectangles: Properties, naming, rotation
Constructing squares and rectangles
Exploration: Distances in rectangles
Breaking rectangles into squares
Advanced constructs: Square with hole, curves
Diagonals in rectangles/squares
Equidistant points: House
Rhombus construction
Key Takeaways for Exams
Circle Basics
Fixed distance from center; use compass.
Rectangle Properties
Opposite sides equal, 90° angles.
Square Properties
All sides equal, 90° angles.
Constructions
Use perpendiculars, arcs for points.
Diagonals
Equal lengths, divide angles.
Equidistant Points
Intersecting arcs/circles.
Key Rules & Properties – Constructions
Important rules for geometric figures and constructions.
Circle Properties
Property
Description
Center
Fixed point P equidistant to all points on curve.
Radius
Distance from center to any point on circle.
Drawing
Fix compass point, rotate pencil.
Rectangle Properties (R)
Property
Description
R1
Opposite sides equal.
R2
All angles 90°.
Naming
Corners in order around boundary.
Diagonals
Equal length; divide opposite angles.
Square Properties (S)
Property
Description
S1
All sides equal.
S2
All angles 90°.
Rotation
Properties unchanged.
Construction Rules
Perpendiculars: Use set square or compass for 90°.
Equidistant Points: Intersect circles/arcs.
Transfer Lengths: Compass without ruler.
Diagonals Divide Angles: Equal in squares.
Concept Cards – Quick Explanations
Circle
Curve equidistant from center; radius fixed.
Compass Use
Draw arcs/circles; transfer lengths.
Rectangle
Opposite sides equal, right angles.
Square
All sides equal, right angles.
Rotation
Properties preserved for squares/rectangles.
Diagonals
Equal in rectangles; bisect angles in squares.
Equidistant Points
Intersection of two circles.
Rhombus
All sides equal, not necessarily square.
Perpendicular
90° line; construct via compass/ruler.
Breaking Rectangles
Into identical squares; length multiple of side.
Examples + Solutions
Example 1: Constructing a Circle
Solution: Fix compass at P with radius 4 cm, rotate pencil.
Example 2: Naming Rectangle
Solution: ABCD, BCDA, etc.; not ABDC.
Example 3: Constructing Square of Side 6 cm
Solution: Draw PQ 6 cm, perpendiculars at P/Q, mark S/R 6 cm, join.
Example 4: Rectangle with Side 5 cm, Diagonal 7 cm
Solution: Draw base, perpendicular, arc from end to intersect.
Example 5: House with Sides 5 cm
Solution: Base BC, arcs from B/C to find A, arc from A.
Example 6: Rectangle Divided into 3 Squares
Solution: Length 3 times breadth.
Example 7: Diagonals Dividing Angles 60°/30°
Solution: Draw base, angle at A, perpendiculars.
Figure it Out Solutions (All Solved)
Section 8.1: Wavy Wave
1. Radius for half circle? Length AX?
Ans. 2 cm. AX = 4 cm.
2. Different length central line.
Ans. Adjust radius to half length.
3. Smaller waves.
Ans. Use smaller radius, symmetric placement.
Section 8.2
Invalid square name?
Ans. PQSR.
1. Draw rectangle + squares on dot paper.
Ans. Leave dot distance diagonally.
2. Identify squares.
Ans. A is square.
Think: Reason without instruments?
Ans. Yes, dot positions for equality/angles.
3. Draw 3 rotated squares/rectangles.
Ans. Verify properties hold.
Section 8.3: Construct
1. Rectangle 4x6 cm.
Ans. Angles 90°, opposites equal.
2. Rectangle 2x10 cm.
Ans. Angles 90°, opposites equal.
3. Possible 4-sided with 90° but opposites unequal?
Ans. No.
Section 8.4
Table for distances.
Ans. Example: 5mm/3cm = 7.4cm, etc.
Same distances table.
Ans. XY=7cm always.
Observe: XY vs AB; shape ABYX.
Ans. XY=AB; ABYX rectangle.
Farthest vs AC/BD.
Ans. Equal to AC/BD.
Construct rectangle into 3 squares.
Ans. Length 3x breadth.
Rectangles not dividable into 2/3 squares.
Ans. e.g., 4x2.5 cm (2); 7x2 cm (3).
4. Square with Hole.
Ans. Center at diagonals intersection.
Section 8.5: Explore
Diagonal equal angles?
Ans. When square.
Section 8.5: Construct
1. Diagonal divides 50°/40°.
Ans. Follow steps with angles.
2. 45°/45°.
Ans. Sides equal (square).
3. Side 4 cm, diagonal 8 cm.
Ans. Construct as per method.
4. Side 3 cm, diagonal 7 cm.
Ans. Similar construction.
Extra Practice Questions (Exam-Ready) – Chapter 8
25+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags
1-Mark Questions (Very Short Answer)
1. What is radius?
Distance from center to point on circle.
2. Rectangle property R1?
Opposite sides equal.
3. Square angles?
All 90°.
4. Invalid square name for PQRS?
PQSR.
5. Tool for circle?
Compass.
2-Mark Questions (Short Answer)
6. Draw circle of radius 4 cm.
Fix compass, rotate.
7. Rectangle diagonals property.
Equal lengths.
8. Equidistant point method.
Intersect arcs.
9. Rotation effect on square.
Remains square.
10. Minimum distance in rectangle exploration.
Equals length AB.
3-Mark Questions (Reasoning)
11. Why use compass for equidistant?
Locates intersection points accurately.
12. Square vs rectangle properties.
Square has all equal sides; rectangle opposites.
13. Diagonals equal angles when?
In square.
14. Construct square side 6 cm steps.
Base, perpendiculars, mark points.
15. Rhombus not square?
Angles not 90°.
4–5 Mark Questions (Application)
16. Construct rectangle 4x6 cm.
Verify properties.
17. House sides 7 cm.
Base, arcs, roof arc.
18. Rectangle into 3 squares.
Length 3x side.
19. Diagonal divides 45°/45°.
Becomes square.
20. Square with curves.
Arcs from midpoints.
Challenge Questions (6+ Marks)
21. Explain rectangle exploration table.
Same distances give constant XY=AB.
22. Construct rhombus.
Equidistant from two points.
23. Square with hole center.
Diagonals intersection.
24. Side 3 cm, diagonal 7 cm rectangle.
Base, perpendicular, arc.
25. Eyes construction hint.
Symmetric arcs with supporting lines.
Common Mistakes & How to Avoid
Mistake 1: Incorrect Compass Radius
Not measuring accurately against ruler.
Avoid: Always check distance tip to pencil.
Mistake 2: Wrong Naming Order
Not following boundary travel.
Avoid: Label in clockwise/counterclockwise sequence.
Mistake 3: Assuming Rotation Changes Properties
Thinking rotated square not square.
Avoid: Verify sides/angles unchanged.
Mistake 4: Trial/Error Without Arcs
Guessing points instead of intersecting.
Avoid: Use circles/arcs for precision.
Mistake 5: Unequal Angles in Constructions
Not ensuring 90° with perpendiculars.
Avoid: Use set square or compass method.
Mistake 6: Ignoring Symmetry
Asymmetric waves/eyes.
Avoid: Estimate and measure placements.
History & Fun Facts
Ancient Origins
Euclid's Elements (300 BC) formalized constructions with ruler/compass.
Ancient Greeks solved problems like squaring circle (impossible with tools).
Real-Life Applications
Architecture: Blueprints use precise constructions.
Art: Symmetry in designs, logos.
Engineering: Bridges, machines require accurate angles.
Navigation: Compass for maps.
Fun Facts
Compass invented in China ~200 BC for feng shui.
Squares/rectangles in pixel art, buildings.
Rhombus in kites, diamonds.
Golden ratio constructions in art (e.g., Parthenon).
Impossible constructions: Trisect angle with ruler/compass.
Did You Know?
Mohammed ibn Musa al-Khwarizmi advanced geometry in algebra.
