CBSE Class 6 Mathematics Playing with Constructions MCQs Set 11

Answer: (c) radius

Explanation:
1. All points of a circle are the same distance from the centre.
2. This fixed distance is called the radius.
3. The diameter is twice the radius, and a chord and an arc are parts of the circle itself.

Teacher's Note:
1. Radius: centre to circle.
2. Diameter: right across the circle through the centre, so it is twice the radius.

Answer: (d) any shape that can be drawn on paper with a pencil

Explanation:
1. In this chapter, a curve means any shape that can be drawn on paper with a pencil.
2. So straight lines, circles and wavy shapes are all curves.
3. Options (a), (b) and (c) each name only one kind, so the full answer is (d).

Teacher's Note:
1. In everyday talk a 'curve' is bent, but here even a straight line counts as a curve.
2. Watch for the word 'only' in the options.

Answer: (a) 3.5 cm

Explanation:
1. The gap between the metal tip and the pencil tip is the radius of the circle drawn.
2. The radius needed is 3.5 cm, so the compass is opened to 3.5 cm.

Teacher's Note:
1. Compass opening = radius, not diameter.
2. 7 cm would be the diameter of this circle.

Answer: (b) 90°

Explanation:
1. Property R2 says that all the angles of a rectangle are 90°.
2. So each corner of a notebook page measures 90°, and it fits the corner of a set square exactly.

Teacher's Note:
1. A rectangle has four right angles.
2. A set square is a quick check for a right angle.

Answer: (d) all four sides are equal and all four angles are 90°

Explanation:
1. A square needs S1 (all sides equal) and S2 (all angles 90°) together.
2. Equal sides alone can give a slanting diamond shape; right angles alone give a rectangle.
3. So only (d) is enough.

Teacher's Note:
1. A square needs both conditions together.
2. Remember the counter-examples: a tilted diamond and a long rectangle.

Answer: (c) KMLN

Explanation:
1. In a correct name the letters follow the order in which we go round the figure, in either direction.
2. LMNK and NKLM go round one way; MLKN goes round the other way. All three are fine.
3. KMLN jumps from K straight across to M, so KMLN is not a correct name.

Teacher's Note:
1. Check that each pair of neighbouring letters is a side of the figure.
2. K and M are opposite corners, so KM is a diagonal, not a side.

Answer: (c) Both Statements are true

Explanation:
1. Instruments give neater, more exact figures than freehand drawing, so Statement I is true.
2. A compass draws circles and arcs (parts of circles), so Statement II is true.
3. Hence both statements are true.

Teacher's Note:
1. Judge each statement on its own first.
2. An arc is a part of a circle.

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

Explanation:
1. Turning the sheet keeps every side length and every angle unchanged, so R is true.
2. The sheet still has four equal sides and four right angles, so it is still a square and A is true.
3. R gives the reason for A, so the answer is (a).

Teacher's Note:
1. A shape's name depends on its sides and angles, not on how it is placed.
2. A tilted square is still a square.

Answer: (d) A is false, but R is true.

Explanation:
1. A four-sided figure with all angles 90° is a rectangle.
2. In a rectangle the opposite sides must be equal (property R1), so A is false.
3. R is a true statement of R1. So the answer is (d).

Teacher's Note:
1. Try to draw such a figure; it cannot be done.
2. R1 and R2 always come together.

Answer: (b) (i)-(q), (ii)-(r), (iii)-(s), (iv)-(p)

Explanation:
1. The centre is the fixed point of a circle: (i)-(q).
2. The radius is the distance from the centre to any point on the circle: (ii)-(r).
3. A corner is where two sides meet: (iii)-(s). A diagonal joins two opposite corners: (iv)-(p).
4. So the answer is (b).

Teacher's Note:
1. Match the easiest pair first, then rule out options.
2. Only option (b) has (i)-(q).

Answer: (c) AC and BD are equal in length

Explanation:
1. The two diagonals of a rectangle are always equal in length, so AC = BD.
2. A diagonal goes right across the frame, so it is longer than any side; AC cannot be equal to AB.
3. So AC and BD are equal in length.

Teacher's Note:
1. Measure both diagonals in your own drawing to see this.
2. A diagonal is always longer than a side of the rectangle.

Answer: (b) 5

Explanation:
1. There is one small square at each corner: 4 squares.
2. The big outer square is one more square.
3. The circle in the middle is not a square. Total = 4 + 1 = 5.

Teacher's Note:
1. Do not forget the big outer square.
2. Count shapes in groups: small ones first, then the big one.

Answer: (d) PQ and RS

Explanation:
1. Going round the rectangle the sides are PQ, QR, RS and SP.
2. Opposite sides lie across from each other: PQ with RS, and QR with SP.
3. Options (a), (b) and (c) are sides that meet at a corner. So the answer is PQ and RS.

Teacher's Note:
1. Two sides sharing a letter meet at a corner, so they are not opposite.
2. Opposite sides have no letter in common.

Answer: (b) a set square

Explanation:
1. A set square has a ready-made right angle, so placing it along the line gives the perpendicular at once.
2. A protractor also works (by marking 90°), but a divider only compares lengths and a ruler alone cannot make a right angle.
3. So the answer is a set square.

Teacher's Note:
1. Set square = ready-made 90° corner.
2. Know the job of each instrument in the geometry box.

Answer: (d) 3 cm

Explanation:
1. X is the mid-point, so AX = 12 ÷ 2 = 6 cm. This is the diameter of the first half circle.
2. The radius is half the diameter: 6 ÷ 2 = 3 cm.

Teacher's Note:
1. Halve twice: once for the mid-point, once for the radius.
2. 6 cm is the diameter, a common wrong answer.

Answer: (a) Only Statement I is true

Explanation:
1. A square has opposite sides equal and all angles 90°, so it satisfies R1 and R2. Statement I is true.
2. A rectangle such as 6 cm × 4 cm does not have all sides equal, so it is not a square. Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. A square is a special rectangle.
2. One counter-example is enough to show Statement II is false.

Answer: (b) Only Statement II is true

Explanation:
1. The side PQ is drawn first; only then can a perpendicular be drawn to it at P. Statement I is false.
2. The compass can copy the length PQ on to the perpendicular at Q to fix R. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. A perpendicular needs a line to stand on.
2. A compass copies lengths without measuring again.

Answer: (c) A is true, but R is false.

Explanation:
1. 15 = 3 × 5, so the rectangle splits into three squares of side 5 cm. A is true.
2. This works only when the length is exactly three times the breadth, so R is false.
3. So the answer is (c).

Teacher's Note:
1. Check 'length = 3 × breadth'.
2. Words like 'every' often make a statement false.

Answer: (d) A is false, but R is true.

Explanation:
1. The sides 7 cm and 4 cm are not equal, so the two parts of the corner angle are not equal. A is false.
2. The parts become 45° each only when all four sides are equal, that is, in a square. R is true.
3. So the answer is (d).

Teacher's Note:
1. 45° and 45° happens only in a square.
2. In a 7 cm × 4 cm rectangle the parts are unequal, though they still add to 90°.

Answer: (c) 9 cm

Explanation:
1. X and Y are the same distance (2 cm) from A and B.
2. So ABYX is a rectangle and XY is opposite to AB.
3. Hence XY = AB = 9 cm, the shortest possible distance between X and Y.

Teacher's Note:
1. Equal distances from A and B give a straight-across line.
2. Opposite sides of a rectangle are equal.

Answer: (d) (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q)

Explanation:
1. A ruler draws a straight line of a given length: (i)-(r).
2. A compass draws circles and arcs: (ii)-(s).
3. A protractor draws and measures angles: (iii)-(p). A divider compares two lengths: (iv)-(q).
4. So the answer is (d).

Teacher's Note:
1. Think of what each tool looks like.
2. A divider has two metal points and no pencil.

Answer: (b) 45°

Explanation:
1. In a square, a diagonal cuts each corner angle of 90° into two equal parts.
2. So x = 90° ÷ 2 = 45°.

Teacher's Note:
1. Square + diagonal = two 45° angles.
2. This is true for every square, big or small.

Answer: (c) 55°

Explanation:
1. The diagonal AC splits the right angle at A into two parts that add up to 90°.
2. So ∠CAD = 90° − 35° = 55°.

Teacher's Note:
1. The two parts of a corner angle of a rectangle always add up to 90°.
2. Subtract the given part from 90°.

Answer: (b) 2 cm

Explanation:
1. The square fits exactly across the 4 cm breadth.
2. Along the length, 8 − 4 = 4 cm is left over.
3. The centres are the same, so this is shared equally: 4 ÷ 2 = 2 cm on each side.

Teacher's Note:
1. Same centre means equal gaps on both ends.
2. Draw a rough diagram to see the two gaps.

Answer: (a) XY can never be less than 12 cm

Explanation:
1. XY is shortest when X and Y are the same distance from A and B; then ABYX is a rectangle and XY = AB = 12 cm.
2. Any other position makes XY slant across, so it gets longer.
3. So XY can never be less than 12 cm.

Teacher's Note:
1. X and Y are on the two opposite 5 cm sides, which are 12 cm apart.
2. The straight-across position gives the least length.

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

Explanation:
1. A diagonal divides the corner angles equally only when all four sides are equal, so such a rectangle is a square. A is true.
2. The two diagonals of every rectangle are equal, so R is also true.
3. But equal diagonals are found in every rectangle, square or not, so R does not explain A. The answer is (b).

Teacher's Note:
1. Ask: does R give the reason for A?
2. Equal diagonals do not single out a square.

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

Explanation:
1. Every point of the arc with centre B is 5 cm from B, so R is true.
2. In the same way, every point of the arc with centre C is 5 cm from C.
3. So the crossing point is 5 cm from both, and A is true because of R. The answer is (a).

Teacher's Note:
1. An arc is a collection of points all at the same distance from its centre.
2. Two arcs find the point at once, with no trial and error.

Answer: (d) Neither Statement is true

Explanation:
1. Small arcs near the expected place are enough; full circles are not needed. Statement I is false.
2. A slanting four-sided figure can have all sides equal without having 90° angles, so it is not a square. Statement II is false.
3. So neither statement is true.

Teacher's Note:
1. Small arcs save time and keep the page clean.
2. Equal sides alone do not make a square; the angles must be 90° too.

Answer: (a) (i)-(r), (ii)-(q), (iii)-(p), (iv)-(s)

Explanation:
1. The two diagonals of a rectangle are equal in length: (i)-(r).
2. A diagonal of a square makes 45° with a side: (ii)-(q).
3. The two parts of a corner angle add up to 90°: (iii)-(p). A rectangle has 2 diagonals: (iv)-(s).
4. So the answer is (a).

Teacher's Note:
1. (i) and (iv) are easy; start with them.
2. Then only one option is left with (i)-(r) and (iv)-(s) and the right middle pairs.

Answer: (a) Draw an arc of radius 7 cm with centre D and mark where it cuts l

Explanation:
1. The corner B lies on l and is 7 cm from D.
2. An arc of radius 7 cm with centre D contains all points 7 cm from D, so it cuts l exactly at B.
3. Option (c) is trial and error, and (b) and (d) use the wrong centre or the wrong line. So the answer is (a).

Teacher's Note:
1. The diagonal DB starts at D, so the arc must be centred at D.
2. Arc + line = exact point, no guessing.