Class 6 Mathematics – NCERT (Ganita Prakash)

Chapter 8: PLAYING WITH CONSTRUCTIONS

STUDY NOTES & KEY PROPERTIES

1. Introduction to Geometric Tools

Compass: Used to draw circles and arcs of a given radius.
Ruler: Used to draw straight lines and measure lengths.
A curve is any shape that can be drawn on paper, including straight lines, circles, and other figures.

2. Circle and Its Parts

Center (P): The fixed point from which all points on the circle are equidistant.
Radius: The distance from the center to any point on the circle.
All points on a circle are at the same distance from the center.

3. Squares and Rectangles

Rectangle Properties:
- Opposite sides are equal.
- All angles are 90°.
Square Properties:
- All sides are equal.
- All angles are 90°.
A square is a special type of rectangle.
Naming of quadrilaterals follows the order of corners around the shape.

4. Constructing Squares and Rectangles

A square can be constructed using a compass and ruler given its side length.
A rectangle can be constructed given:
- Lengths of two adjacent sides, or
- One side and the diagonal length.

5. Diagonals of Rectangles and Squares

Diagonals of a rectangle are equal in length.
In a square, diagonals also bisect the angles into two equal parts (45° each).

6. Points Equidistant from Two Given Points

Using a compass, we can find points that are equidistant from two given points by drawing arcs or circles of equal radius from both points.

7. Practical Construction Tips

Draw a rough diagram before starting construction.
Use a compass to transfer lengths without a ruler.
Light construction lines can help in locating points accurately.

❓ QUESTION BANK

A. Multiple Choice Questions (20 Questions)

In a rectangle, how many pairs of opposite sides are equal?
a) 1
b) 2
c) 3
d) 4
If all sides of a quadrilateral are equal and all angles are 90°, it is a:
a) Rectangle
b) Rhombus
c) Square
d) Parallelogram
How many right angles does a square have?
a) 1
b) 2
c) 3
d) 4
Which of the following is not a valid name for a rectangle with vertices A, B, C, D?
a) ABCD
b) BCDA
c) ACBD
d) CDAB
Which property is true for both squares and rectangles?
a) All sides equal
b) Diagonals equal
c) All angles 90°
d) Opposite sides parallel
If a diagonal of a rectangle divides an angle into 60° and 30°, the other angles are:
a) 60° and 30°
b) 90° each
c) 120° and 60°
d) 45° each
To construct a square of side 5 cm, which step comes first?
a) Draw a perpendicular
b) Draw a line of 5 cm
c) Draw a diagonal
d) Draw a circle
A rectangle with sides 4 cm and 6 cm has a perimeter of:
a) 10 cm
b) 20 cm
c) 24 cm
d) 30 cm
Which shape can be divided into two identical squares?
a) Any rectangle
b) Rectangle with sides in ratio 1:2
c) Rectangle with sides in ratio 2:3
d) Only a square
How many arcs are drawn to locate a point equidistant from two given points?
a) 1
b) 2
c) 3
d) 4
In a square, each diagonal divides the opposite angles into:
a) Two equal parts
b) Unequal parts
c) 60° and 30°
d) 90° each
Which of these is a curve?
a) Only circle
b) Only straight line
c) Both straight line and circle
d) Only triangle
If AB = 8 cm in a rectangle, then CD =
a) 4 cm
b) 8 cm
c) 12 cm
d) 16 cm
A rotated square is still a square because:
a) Sides change
b) Angles change
c) Both sides and angles remain same
d) It becomes a rectangle
Which instrument is used to draw a perpendicular line?
a) Only compass
b) Only ruler
c) Compass and ruler
d) Protractor
A rectangle with one side 5 cm and diagonal 7 cm will have the other side approximately:
a) 4.9 cm
b) 5 cm
c) 6 cm
d) 8.6 cm
The number of diagonals in a rectangle is:
a) 1
b) 2
c) 3
d) 4

B. Assertion & Reasoning (20 Questions)

Directions: Choose the correct option:

(a) Both A and R are true, and R is the correct explanation of A.

(b) Both A and R are true, but R is not the correct explanation of A.

(d) A is false, but R is true

Assertion: All squares are rectangles.
Reason: A square has all angles 90° and opposite sides equal.
Assertion: A compass can be used to draw a circle of radius 4 cm.
Reason: A compass has a pencil and a pointed tip.
Assertion: A rotated square remains a square.
Reason: Rotation changes the side lengths.
Assertion: Diagonals of a rectangle are equal.
Reason: Diagonals of a square are not equal.
Assertion: A rectangle can be named in 8 different ways.
Reason: The vertices can be taken in any order.
Assertion: To find a point equidistant from two points, we draw two arcs.
Reason: The intersection of two circles gives points at equal distance from both centers.
Assertion: A square can be divided into two identical rectangles.
Reason: A square has all sides equal.
Assertion: In a rectangle, if one diagonal divides an angle into 45° and 45°, the rectangle is a square.
Reason: In a square, diagonals bisect the angles equally.
Assertion: A circle is a curve.
Reason: A straight line is also a curve.
Assertion: A rhombus with all angles 90° is a square.
Reason: A rhombus has all sides equal.
Assertion: Using a compass, we can transfer lengths without a ruler.
Reason: A compass can measure angles.
Assertion: A rectangle with sides 3 cm and 4 cm has a diagonal of 5 cm.
Reason: For a rectangle, diagonal = √(length² + breadth²).
Assertion: In the “House” construction, point A is found using two circles.
Reason: Point A is 5 cm from both B and C.
Assertion: A quadrilateral with all sides equal is a square.
Reason: A rhombus also has all sides equal.
Assertion: A rectangle cannot be divided into 3 identical squares if the sides are not in ratio 1:3.
Reason: For 3 identical squares, the longer side must be 3 times the shorter side.
A square satisfies the following two properties:
Assertion (A): S1) All the sides are equal,.
Reason (R): and S2) All the angles are 90°.
Assertion: A square of side 6 cm has a diagonal of about 8.5 cm.
Reason: Diagonal of a square = side × √2.
Assertion: In a rectangle, diagonals bisect each other.
Reason: Diagonals of a rectangle are perpendicular.
Assertion: To construct a rectangle given one side and a diagonal, we use a circle.
Reason: The third vertex lies on the intersection of a circle and a perpendicular line.
Assertion: A rotated rectangle is still a rectangle.
Reason: Rotation does not change lengths and angles.

C. True/False (10 Questions)

A compass can only draw full circles, not arcs.
All rectangles are squares.
A square has 4 lines of symmetry.
The diagonals of a square are equal.
A quadrilateral with all angles 90° must be a square.
A rectangle can be constructed if only one side is known.
In a rectangle, opposite sides are parallel.
A rhombus is a square if one angle is 90°.

D. Short Answer Type I (2 Marks each – 15 Questions)

Write two properties of a rectangle.
Draw a rough sketch of a square PQRS of side 5 cm.
How many different ways can you name a rectangle with vertices W, X, Y, Z?
If a rectangle has length 8 cm and breadth 6 cm, what is the length of its diagonal?
How do you draw a perpendicular to a line using a compass?
What is the shape of the curve obtained by keeping the compass tip fixed and moving the pencil?
Write one similarity and one difference between a square and a rectangle.
In a rectangle ABCD, if AB = 7 cm and BC = 5 cm, what are the lengths of CD and AD?
What is the minimum number of measurements needed to construct a square?
If a diagonal of a square is 10 cm, what is the side length?
Can a rectangle be divided into two identical squares? If yes, give an example of side lengths.
What is the purpose of drawing light construction lines?

E. Short Answer Type II (3 Marks each – 10 Questions)

Construct a rectangle with sides 5 cm and 3 cm. Verify its properties.
Explain how to locate a point that is 4 cm from point P and 4 cm from point Q.
Draw a square of side 4 cm without using a protractor.
Divide a rectangle of sides 9 cm and 3 cm into three identical squares. Show construction steps.
In a rectangle, one diagonal divides an angle into 55° and 35°. What are the other angles?
Construct a rectangle with one side 6 cm and diagonal 10 cm.
How will you draw the “Wavy Wave” pattern using a compass?
Using a compass, bisect a line segment of length 8 cm.

F. Long Answer Type (5 Marks each – 10 Questions)

Construct a square of side 6 cm. Measure its diagonals and verify they are equal.
Construct a rectangle ABCD with AB = 8 cm and BC = 5 cm. Draw its diagonals and measure the angles they make with the sides.
Construct the “House” figure with all sides 5 cm. Show all construction arcs.
Construct a rectangle that can be divided into two identical squares. Explain your steps.
Construct a rectangle with one side 7 cm and a diagonal 9 cm. Verify rectangle properties.
Draw a square with 8 cm side. Inside it, draw a circle touching all four sides.
Construct a rectangle where one diagonal divides opposite angles into 60° and 30°.
Construct a “Wavy Wave” with central line 10 cm and half-circle waves.
Draw two identical “Eyes” using compass construction.
Construct a square with a circular hole at the center such that the circle touches all sides.

G. Case-Based Questions (5 Cases, each with 4 Sub-Questions)

Case 1: A rectangle has vertices A, B, C, D. AB = 6 cm, BC = 4 cm. Diagonals AC and BD intersect at O.

What is the length of CD?
a) 4 cm
b) 6 cm
c) 10 cm
d) 8 cm
What is the length of diagonal AC?
a) 7.2 cm
b) 10 cm
c) 8.5 cm
d) 9 cm
If ∠CAB = 30°, then ∠ACB =
a) 30°
b) 60°
c) 90°
d) 120°
How many pairs of equal triangles are formed by the diagonals?
a) 2
b) 4
c) 6
d) 8

Case 2: A square sheet of side 10 cm is rotated to look like a diamond.

Is it still a square?
a) Yes
b) No
What is the length of each side after rotation?
a) Changes
b) Remains 10 cm
c) Becomes 5 cm
d) Doubles
What is the angle between two adjacent sides after rotation?
a) 60°
b) 90°
c) 120°
d) 45°
How many lines of symmetry does it have now?
a) 1
b) 2
c) 4
d) 0

Case 3: In the “House” construction, all edges are 5 cm.

How many arcs are needed to locate point A?
a) 1
b) 2
c) 3
d) 4
What is the shape of the roof?
a) Triangle
b) Square
c) Circular arc
d) Rectangle
Which tool is essential for this construction?
a) Protractor
b) Compass
c) Set-square
d) Divider
The base BC is of length:
a) 5 cm
b) 10 cm
c) 15 cm
d) 20 cm

Case 4: A rectangle is divided into 3 identical squares.

If the shorter side of rectangle is 4 cm, the longer side is:
a) 8 cm
b) 12 cm
c) 16 cm
d) 20 cm
How many squares are formed in total?
a) 2
b) 3
c) 4
d) 6
What is the perimeter of each small square?
a) 8 cm
b) 12 cm
c) 16 cm
d) 20 cm
Can this rectangle be a square?
a) Yes
b) No

🌈⚡Key To Enjoy Learning Maths☘🌈

Thursday, December 11, 2025