CBSE Class 7 Mathematics Constructions and Tilings MCQs Set 11

Answer: (b) bisection

Explanation:
1. Splitting a segment or any object into two identical parts is called bisection.
2. Construction means drawing exactly; reflection is a mirror image; tiling is covering a region.
3. None of these other words means cutting into two equal halves.

Teacher's Note:
1. 'Bi' means two, as in bicycle.
2. The point that bisects a segment is its midpoint.

Answer: (c) perpendicular bisector

Explanation:
1. The line passes through the midpoint (bisects) and makes 90° with the segment (perpendicular).
2. Both ideas together give the name perpendicular bisector.
3. An angle bisector splits an angle, and a transversal cuts across two other lines.

Teacher's Note:
1. The name itself lists both conditions.
2. Midpoint plus 90° means perpendicular bisector.

Answer: (d) an unmarked ruler and a compass

Explanation:
1. The chapter uses only the simplest instruments: an unmarked ruler and a compass.
2. The ruler draws straight lines and the compass draws arcs and copies lengths.
3. A marked scale is used only when a length in standard units, such as 4 cm, is needed.

Teacher's Note:
1. No protractor is used for these constructions.
2. Ruler-and-compass constructions are exact.

Answer: (a) the perpendicular bisector of XY

Explanation:
1. Equal radii give AX = AY and BX = BY.
2. So A and B are each equidistant from X and Y, and both lie on the perpendicular bisector of XY.
3. Hence AB is the perpendicular bisector of XY; it meets XY at its midpoint O at 90°.

Teacher's Note:
1. Points equidistant from X and Y always lie on the perpendicular bisector.
2. Two such points fix the line.

Answer: (c) tiling

Explanation:
1. Covering a region completely with shapes, leaving no gaps and making no overlaps, is called tiling.
2. A tiled floor is the everyday example.
3. Bisection, congruence and symmetry describe other ideas.

Teacher's Note:
1. Two conditions: no gaps, no overlaps.
2. Think of a floor laid with tiles.

Answer: (b) Regular hexagon

Explanation:
1. Copies tile the plane only if their angles fill exactly 360° around a point.
2. Each angle of a regular hexagon is 120°, and 3 × 120° = 360°.
3. The pentagon (108°), heptagon and octagon (135°) do not divide 360° exactly. Answer: Regular hexagon.

Teacher's Note:
1. Beehive cells are regular hexagons.
2. Check whether the angle divides 360° exactly.

Answer: (c) 45°

Explanation:
1. The angles around a point add up to 360°.
2. Eight equal angles: each angle = 360° ÷ 8.
3. So each angle is 45°.

Teacher's Note:
1. This is the angle of the eight-petalled design.
2. 45° is got by bisecting 90°.

Answer: (b) an equilateral triangle

Explanation:
1. An arc from A cuts the ray at B; an arc of the same radius from B cuts the first arc at C.
2. Then AB = AC = BC, so △ABC is an equilateral triangle.
3. Each angle of an equilateral triangle is 60°, so ∠CAX = 60°.

Teacher's Note:
1. Keep the same compass opening for both arcs.
2. Three equal sides give three 60° angles.

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

Explanation:
1. Angles fit around a point with no gap and no overlap exactly when they add up to 360°.
2. Six angles of 60° give 6 × 60° = 360°, so A is true.
3. R is true and is exactly the reason for A.

Teacher's Note:
1. This is how a regular hexagon is built.
2. Always check the total against 360°.

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

Explanation:
1. The seven tangram pieces are cut from a square, not a circle, so A is false.
2. Tangrams did originate in China, so R is true.
3. Hence A is false but R is true.

Teacher's Note:
1. Tangram pieces have straight edges only.
2. Seven pieces, one square.

Answer: (c) Both statements are true

Explanation:
1. Statement I is the definition of tiling, so it is true.
2. A 2 × 1 tile may be turned, so it can lie vertically or horizontally. Statement II is true.
3. Hence both statements are true.

Teacher's Note:
1. Turning a tile is allowed.
2. Remember the two conditions of tiling.

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

Explanation:
1. Bisecting 90° gives 45°, so (i)-(r); bisecting 60° gives 30°, so (ii)-(s).
2. Bisecting 30° gives 15°, so (iii)-(p); 60° + 60° = 120°, so (iv)-(q).
3. Correct matching: (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q).

Teacher's Note:
1. Bisecting halves the angle.
2. Placing angles side by side adds them.

Answer: (b) 24

Explanation:
1. Number of unit squares = 6 × 8 = 48.
2. Each 2 × 1 tile covers 2 unit squares.
3. Number of tiles = 48 ÷ 2 = 24.

Teacher's Note:
1. Count the squares first, then divide by 2.
2. 48 is the number of squares, not tiles.

Answer: (b) 35 is odd, while tiles covering 2 squares each always cover an even number of squares

Explanation:
1. Each tile covers exactly 2 squares, so any number of tiles covers an even number of squares.
2. 35 is odd, so one square is always left over.
3. So the reason is: 35 is odd, while tiles always cover an even number of squares.

Teacher's Note:
1. Option (a) is false anyway: 35 is a multiple of 5.
2. Odd number of squares means no tiling.

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

Explanation:
1. The two arcs above use one radius, so their meeting point A is equidistant from X and Y.
2. The two arcs below use one (possibly different) radius, so B is also equidistant from X and Y.
3. Both points lie on the perpendicular bisector and two points fix a line. A and R are true and R explains A.

Teacher's Note:
1. Each pair must use one radius, but the two pairs need not match.
2. Two points fix a line.

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

Explanation:
1. Marking X and Y with OX = OY makes O the midpoint of XY, so O lies on its perpendicular bisector. R is true.
2. Only one more point of that line is needed, so a single pair of intersecting arcs is enough.
3. Hence A is false but R is true.

Teacher's Note:
1. O is already one of the two points needed.
2. The word 'always' makes A false.

Answer: (a) Only Statement I is true

Explanation:
1. With OA = OB, AC = BC and OC common, the triangles are congruent by SSS. Statement I is true.
2. Parallel lines need equal corresponding angles, not unequal ones. Statement II is false.
3. Hence only Statement I is true.

Teacher's Note:
1. OC is the common third side.
2. Equal corresponding angles give parallel lines.

Answer: (d) Neither statement is true

Explanation:
1. A regular polygon has all sides equal and all angles equal, so Statement I is false.
2. Each angle of a regular hexagon is 60° + 60° = 120°, so Statement II is false.
3. Hence neither statement is true.

Teacher's Note:
1. Regular means equal sides and equal angles.
2. 60° is the angle of each triangle, not of the hexagon.

Answer: (c) 70°

Explanation:
1. The angles around a point add up to 360°.
2. Marked angles: 80° + 50° + 95° + 65° = 290°.
3. Unmarked angle = 360° − 290° = 70°.

Teacher's Note:
1. Add the known angles and subtract from 360°.
2. This is the gap-angle idea of the chapter.

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

Explanation:
1. The pointed arch uses two equal segments with midpoints marked: (i)-(p).
2. The trefoil arch needs equal angles at A and D with AB = CD: (ii)-(s).
3. The eight-petalled design uses eight 45° angles: (iii)-(q); the hexagon uses six equilateral triangles: (iv)-(r). Answer: (a).

Teacher's Note:
1. Link each figure to its supporting lines.
2. Eight petals means 45°; hexagon means 60° triangles.

Answer: (a) SSS

Explanation:
1. Arcs of the same radius make two sides of one triangle equal to two sides of the other.
2. The chord length transferred with the compass makes the third sides equal.
3. Three pairs of equal sides: the triangles are congruent by SSS.

Teacher's Note:
1. No angle is measured; only lengths are copied.
2. Three equal sides means SSS.

Answer: (c) a transversal

Explanation:
1. The line l cuts m and will also cut the new line through B.
2. A line that cuts two lines is a transversal.
3. The corresponding angle at A is copied at B to get the parallel line.

Teacher's Note:
1. Corresponding angles need a transversal.
2. l need not be perpendicular to m.

Answer: (d) 120°

Explanation:
1. At each vertex of the hexagon, the angles of two triangles meet.
2. Each triangle angle is 60°.
3. So each hexagon angle = 60° + 60° = 120°.

Teacher's Note:
1. Look at a vertex on the outside, not the centre.
2. 108° is the angle of a regular pentagon.

Answer: (b) a rope

Explanation:
1. The Sulba-Sutras dealt with laying out fire altars on the ground.
2. They used a rope: swung about a peg it draws arcs, and stretched tight it gives a straight line.
3. So the rope did the work of both the compass and the ruler.

Teacher's Note:
1. Sulba means rope or cord.
2. These are texts of the Vedic period.

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

Explanation:
1. Shapes fit around a point only if their angles add up to exactly 360°.
2. 3 × 108° = 324° leaves a gap and 4 × 108° = 432° overlaps, so 108° does not divide 360°.
3. So A is true, R is true, and R correctly explains A.

Teacher's Note:
1. 360 ÷ 108 is not a whole number.
2. The gap left by three pentagons is 36°.

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

Explanation:
1. The grid has 7 × 9 = 63 unit squares, an odd number.
2. Tiles always cover an even number of squares, so the grid cannot be tiled. A is false.
3. R is a true necessary condition. Hence A is false but R is true.

Teacher's Note:
1. Both 7 and 9 are odd, so the product is odd.
2. Even count is needed, though not always enough.

Answer: (c) cannot be tiled, because the colouring gives 4 squares of one colour and only 2 of the other

Explanation:
1. In a chessboard colouring, every 2 × 1 tile covers one black and one white square.
2. Colouring the staircase gives 4 squares of one colour (the three diagonal squares and the bottom-left one) and 2 of the other.
3. 4 ≠ 2, so it cannot be tiled, because the colouring gives 4 squares of one colour and only 2 of the other.

Teacher's Note:
1. 6 is even, so counting alone does not decide.
2. Equal colours are needed for a tiling.

Answer: (c) 90°

Explanation:
1. Two octagon angles: 135° + 135° = 270°.
2. The angles at the point must total 360°.
3. Remaining angle = 360° − 270° = 90°, the angle of a square.

Teacher's Note:
1. This octagon-square tiling is common on floors.
2. Subtract from 360° to find the gap.

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

Explanation:
1. Hexagon: 3 × 120° = 360°, so (i)-(p). Square: 4 × 90° = 360°, so (ii)-(s).
2. Triangle: 6 × 60° = 360°, so (iii)-(q). Pentagon: 108° does not divide 360°, so (iv)-(r).
3. Correct matching: (i)-(p), (ii)-(s), (iii)-(q), (iv)-(r).

Teacher's Note:
1. Divide 360° by each angle.
2. Only the pentagon fails.

Answer: (b) Only Statement II is true

Explanation:
1. Bisecting 60° gives 30°, and bisecting 30° gives 15°, so Statement II is true.
2. Repeated bisection of 60° gives 30°, 15°, 7.5°, …, each smaller than the last, so 65.5° is never reached. Statement I is false.
3. Hence only Statement II is true.

Teacher's Note:
1. Bisection only makes angles smaller.
2. 65.5° is even larger than 60°.