Playing with Constructions — Hard

Q1. About a circle with centre P and radius 4 cm, consider: I. Every point on the circle is exactly 4 cm from P. II. Any two points on the circle are always 4 cm apart. III. Points inside the circle are less than 4 cm from P. Which are correct?

• A. I and II only
• B. I and III only
• C. II and III only
• D. All three

Q2. Assertion (A): A compass can be used to construct a square without using a protractor. Reason (R): A compass can copy the length of one side onto another line, replacing the protractor's role of fixing the side length.

• A. Both A and R are true, but R is NOT the correct explanation of A.
• B. Both A and R are true, and R is the correct explanation of A.
• C. A is true, R is false.
• D. A is false, R is true.

Q3. After STEP 6 of the square PQRS construction, a student measures RS and finds it is 6.3 cm instead of 6 cm. The most likely error is

• A. The base PQ was drawn 6.3 cm long
• B. The perpendicular at P or Q was not exactly 90°, so R and S shifted
• C. The compass was set to 6 cm correctly but a different ruler was used
• D. The protractor was upside down throughout

Q4. In a rectangle ABCD, one diagonal divides angle A into 60° and 30°. Consider: I. Triangle ABC has a 60° angle at A. II. Side BC is shorter than side AB. III. The same diagonal divides angle C into 60° and 30° in the same order.

• A. II and III only
• B. I and II only
• C. I and III only
• D. All three

Q5. Assertion (A): If a diagonal of a rectangle divides one corner angle into two equal 45° parts, the rectangle is a square. Reason (R): A diagonal of a square joins two opposite right-angle corners and lies along the line of symmetry between two equal sides.

• A. A is true, R is false.
• B. Both A and R are true, but R is NOT the correct explanation of A.
• C. Both A and R are true, and R is the correct explanation of A.
• D. A is false, R is true.

Q6. Asha constructs a rectangle with side DC = 5 cm and diagonal DB = 7 cm. After the perpendicular l at C, she draws a circle of radius 7 cm centred at D. The circle meets l at two points B and B'. The correct vertex B is

• A. Always the upper intersection, irrespective of side
• B. The point on l that lies on the same side as the intended rectangle
• C. The midpoint of the segment BB'
• D. Neither — a fresh circle must be drawn

Q7. About finding A which is 5 cm from B and from C: I. Full 5 cm circles at B and C are needed. II. Two 5 cm arcs — one from B, one from C — are enough. III. One arc from B alone fixes A. Which are correct?

• A. II only
• B. I only
• C. I and II only
• D. II and III only

Q8. In the House construction, after locating A, the curved roof line from B to C is drawn by

• A. Setting the compass to 5 cm and drawing an arc centred at A that touches both B and C
• B. Joining B and C by a straight line and shading the area above it
• C. Drawing a half circle of radius 2.5 cm on BC
• D. Drawing an arc of radius 10 cm centred at the midpoint of BC

Q9. A 4 cm × 2.5 cm rectangle is to be divided into two identical squares by a single straight cut parallel to its shorter side. Is this possible?

• A. Yes, by cutting at the 1 cm mark from the shorter side
• B. Yes, by cutting at the 2 cm mark, giving two 2 cm × 2.5 cm squares
• C. No, because the cut would have to give squares of side 2.5 cm, but only 4 cm of length is available
• D. No, because no cut parallel to the shorter side can ever give two squares

Q10. Statements about drawing an arc instead of a full circle for the House construction: I. The arc must have radius 5 cm. II. The arc's centre depends on which vertex is being located. III. Drawing only the arc removes the need to also draw the construction's straight lines.

• A. All three
• B. II and III only
• C. I and III only
• D. I and II only

Q11. For the 'Square with Curves' (side 8 cm), Ravi wants each of the four arcs to bulge inward by the same amount. Where should he place the tip of the compass for each arc, and what radius is reasonable?

• A. On a point outside the square, opposite each side, all four points at equal distance and with equal radius
• B. On the midpoint of each side, radius 4 cm
• C. On each corner of the square, radius 8 cm
• D. On the centre of the square, radius equal to the diagonal

Q12. Assertion (A): Every square is a rectangle. Reason (R): A square satisfies both rectangle properties — opposite sides equal and all angles 90° — while in addition all four sides are equal.

• A. A is true, R is false.
• B. Both A and R are true, but R is NOT the correct explanation of A.
• C. Both A and R are true, and R is the correct explanation of A.
• D. A is false, R is true.

Q13. Two methods locate S on the perpendicular at P during a 6 cm square build. M: ruler-mark PS = 6 cm. N: 6 cm compass arc from P on the perpendicular. The key advantage of N over M is

• A. N is faster because rulers cannot reach the perpendicular
• B. N transfers the side length without re-measuring, so the same compass setting can later mark R from Q
• C. N produces a longer side than M
• D. N does not need the perpendicular to be drawn first

Q14. Consider these names for the rectangle with corners labelled A, B, C, D in clockwise order: I. ABCD. II. ADCB. III. BADC.

• A. I is valid only
• B. I, II and III are all valid
• C. I and II are valid; III is not
• D. Only III is valid

Q15. While constructing the House, Aman places his compass tip at point D instead of B for the first arc that should locate A. The most likely outcome is

• A. The point he marks will still be 5 cm from B and C
• B. The point will lie exactly on line BC at its midpoint
• C. The arc will not pass through C at all, so he cannot find a valid A from this arc alone
• D. The rectangle base will collapse