NCERT MCQ Solutions for Class 6 Maths Chapter 8 Playing with Constructions

Every marked point is the same distance from P. These points form a circle, with P as its centre.
Radius = distance from the centre to any point on the circle = 4 cm

A radius joins the centre of a circle to a point on the circle. Since P is the centre and Q lies on the circle, PQ is a radius. Every radius of this circle has the same length.

The compass tip stays at the centre while the pencil travels around it. The opening of the compass is the distance from the centre to the circle.
Required compass opening = radius = 4 cm

The two identical semicircles use equal halves of AB.
Diameter of each semicircle = 8 ÷ 2 = 4 cm
Radius = 4 ÷ 2 = 2 cm
So the compass opening should be 2 cm.

The compass opening determines the radius. Moving the compass to a new centre changes the location of the arc, but keeping the opening unchanged preserves its radius. Equal-sized portions of these circles give identical arcs.

Compass is primarily used to draw circles.

The sum of the four angles of a rectangle is 360°.

A rectangle has two pairs of equal opposite sides. Its four corners are right angles, each measuring 90°. Adjacent sides need not be equal, so a rectangle can be longer in one direction than the other.

A square must satisfy both conditions: equal side lengths and right angles.
All four sides are equal.
Each angle = 90°
Checking only the side lengths is not enough, because an equal-sided figure may have slanting corners.

A valid name follows consecutive corners around the boundary, in either direction. In PQSR, the move from Q to S jumps across a diagonal instead of following a side. The other names follow the boundary correctly.

Rotation changes the direction in which the square is placed, not its measurements. Its four sides remain equal and its four angles remain 90°. Since both square properties still hold, the rotated figure is a square.

Equal sides alone do not make a square. A square also needs four right angles. An equal-sided figure with a slanting corner can exist, but it fails the angle condition required for a square.

All sides are equal is NOT a property of a rectangle.

A rectangle can be constructed if we know one side and one diagonal.

All sides of a square have the same length.
PS = PQ = 6 cm
Placing S on the perpendicular gives a 90° angle at P, while choosing PS = 6 cm gives the required equal adjacent sides.

Opposite sides of a rectangle are equal.
CD = AB = 4 cm
DA = BC = 6 cm
The two lengths alternate as we go around the rectangle, and every corner remains a right angle.

A four-sided figure with four right angles is a rectangle. Opposite sides of a rectangle have equal lengths. Rotating or resizing the figure cannot remove this property while keeping all four angles equal to 90°.

The compass can be opened to the distance AF. Without changing that opening, an arc from A marks a point B at the same distance.
AB = AF
No numerical measurement of the length is needed.

Matching tick marks show equal lengths. They record a relationship between sides without giving a particular measurement. A single tick does not mean 1 cm; all equally marked sides could have any common length.

The curve obtained when all points are at equal distance from a center is the shape of circle.

The method used to divide a rectangle into identical squares is Drawing parallel lines.

X and Y are the same distance along the two opposite sides, so ABYX is a rectangle. Its opposite sides AB and XY are equal.
XY = AB = 7 cm

The points are nearest when they are directly opposite each other, at equal distances from A and B. Then XY runs straight across the rectangle.
Minimum XY = AB = 7 cm
A slanting connection is longer.

The separation is greatest when the points occupy opposite corners. With X at A and Y at C, XY is diagonal AC. The other greatest case is X at D and Y at B, giving diagonal DB.

The longer side consists of two square sides placed end to end.
Longer side = 4 + 4 = 8 cm
The shorter side remains 4 cm, so the completed rectangle measures 8 cm by 4 cm.

Each square has side 4 cm because that is the width of the row. Three such sides form the longer side.
Longer side = 3 × 4 = 12 cm
The rectangle is 12 cm by 4 cm.

This property is true for both squares and rectangles that Opposite sides are equal and all angles are 90°.

A diagonal of a rectangle divide rectangle in two right angled triangles. In a right angle triangle
(Hypotenuse)² = (height)² + (base)²
(10)² = (6)² + (base)²
(base)² = (10)² – (6)² = 100 – 36 = 64
Or base = √64 = 8
Hence, length of other side is 8 cm.

The square fills the 4 cm height of the rectangle. Along the 8 cm length, the unused distance is shared equally at the two ends.
Total gap = 8 − 4 = 4 cm
Gap at each end = 4 ÷ 2 = 2 cm

A diagonal joins opposite corners of a rectangle. P is opposite R, and Q is opposite S.
The diagonals are PR and QS.
The other listed segments join neighbouring corners and are sides.

The two diagonals of a rectangle have equal lengths. Both join opposite corners across the same rectangle.
Other diagonal = first diagonal = 7 cm
This equality also holds when the rectangle is rotated.

Each corner of a rectangle measures 90°. The diagonal divides that corner into two smaller angles.
Other angle = 90° − 50° = 40°
The check is 50° + 40° = 90°.

When a diagonal divides these right angles into equal halves, the rectangle has equal adjacent sides. Its opposite sides are already equal, so all four sides are equal. With four right angles, the rectangle is a square.

B must satisfy two conditions: it lies on the perpendicular at C and is 7 cm from D. Every point on the arc centred at D with radius 7 cm is that distance from D. Its intersection locates B.

The required point is where the perpendicular meets the circle. An arc covering that location contains the same intersection as the full circle. Its points still have the chosen distance from the centre, so the construction remains valid.

At D, side DA must be perpendicular to DC. At B, side BA must be perpendicular to BC. The intersection of these two perpendiculars gives A and completes the required right-angled figure.

A point on the first arc is 5 cm from B, and a point on the second arc is 5 cm from C. Their intersection satisfies both distances.
AB = AC = 5 cm
Choosing the intersection above BC places the roof correctly.

Both B and C are 5 cm from A. Therefore, they lie on the same circle centred at A.
Required radius = AB = AC = 5 cm
Keeping the compass tip at A allows an arc to pass through both points.

A square rotated at an angle is still a square because the angles remain 90° and all sides remain equal.

Ruler is the best instrument for measuring the length of a diagonal.