CBSE Class 7 Mathematics Constructions and Tilings Worksheet Set 01

Very Short Answer Questions

Question 1. What is meant by the bisection of a line segment?
Answer:
Bisection of a line segment means dividing it into two identical (equal) parts; the point of division is its midpoint.

Teacher's Note:
1. Bisect means cut into two equal halves.
2. Mention the midpoint.

Question 2. Define the perpendicular bisector of a line segment.
Answer:
It is the line that passes through the midpoint of the segment and is perpendicular to it (makes 90° with it).

Teacher's Note:
1. Two conditions: midpoint and 90°.
2. Each segment has exactly one such line.

Question 3. Name the two instruments that are used for all the geometric constructions of this chapter.
Answer:
An unmarked ruler and a compass. A marked scale is used only when a length in standard units, such as 4 cm, is needed.

Teacher's Note:
1. The protractor is not used.
2. The compass draws arcs and copies lengths.

Question 4. In the ‘Eyes’ construction, the centres A and B are chosen on either side of the supporting line XY. Write the equality that the four distances AX, AY, BX and BY must satisfy.
Answer:
AX = AY = BX = BY. These equal distances make the upper and lower arcs of equal radius, so the eye is symmetrical about XY.

Teacher's Note:
1. All four distances are equal.
2. Equal radii give a symmetric eye.

Question 5. The arcs of equal radius drawn from X and Y meet at A above XY and at B below XY. What is the measure of the angle between AB and XY?
Answer:
90°. AB is the perpendicular bisector of XY, so it cuts XY at right angles at its midpoint.

Teacher's Note:
1. A and B are both equidistant from X and Y.
2. Perpendicular means 90°.

Question 6. What is meant by tiling a region?
Answer:
Tiling a region means covering it completely with a set of shapes, leaving no gaps and making no overlaps.

Teacher's Note:
1. Remember both conditions.
2. A tiled floor is an example.

Question 7. How many pieces does a tangram set have, and which shape are they cut from?
Answer:
A tangram has 7 pieces, obtained by dividing a square.

Teacher's Note:
1. Tangrams originated in China.
2. The pieces can be rearranged into many figures.

Question 8. What is a regular polygon? Name two regular polygons that you have already constructed in earlier classes.
Answer:
A regular polygon has all its sides equal and all its angles equal.
Examples: the equilateral triangle (3 sides) and the square (4 sides).

Teacher's Note:
1. Equal sides alone are not enough.
2. A rhombus has equal sides but is not regular.

Question 9. Name the ancient Indian texts that contain exact methods for constructing perpendicular bisectors, and state the tool used in them in place of a compass.
Answer:
The Sulba-Sutras. They used a rope, which can be swung about a peg to draw arcs and stretched to give a straight line.

Teacher's Note:
1. These texts deal with fire altars.
2. The rope replaces both compass and ruler.

Question 10. Which angle is obtained when a 90° angle is bisected?
Answer:
45°, because 90° ÷ 2 = 45°.

Teacher's Note:
1. Bisecting halves an angle.
2. 45° is used in the eight-petalled design.

Question 11. Name the Dutch artist whose famous works tile the plane with animal shapes.
Answer:
M. C. Escher (1898–1972).

Teacher's Note:
1. His tiles are not polygons yet still leave no gaps.
2. Tiling is still studied by mathematicians today.

Question 12. Which triangle do we construct in order to get a 60° angle with a ruler and a compass?
Answer:
An equilateral triangle. Each of its angles is 60°, so one of its angles gives the 60° angle.

Teacher's Note:
1. Keep the compass opening fixed.
2. All three sides equal the radius.

Question 13. How many perpendicular bisectors does a given line segment have?
Answer:
Exactly one. A segment has only one midpoint and only one line through it at 90° to the segment.

Teacher's Note:
1. All equidistant points lie on this one line.
2. Two such points fix it.

Question 14. State the property of points that makes the ruler-and-compass construction of a perpendicular bisector work.
Answer:
Any point that is at the same distance from both endpoints of the segment lies on its perpendicular bisector. Joining two such points gives the bisector.

Teacher's Note:
1. The arcs find equidistant points.
2. Two points are enough to draw a line.

Question 15. In the construction of the bisector of ∠AOB, the points are taken so that OA = OB and AC = BC. Which congruence condition then shows that OC is the bisector?
Answer:
The SSS condition: OA = OB, AC = BC and OC is common, so △OAC ≅ △OBC and ∠AOC = ∠BOC.

Teacher's Note:
1. The third side OC is shared.
2. Congruent triangles give equal angles.

Question 16. While copying an angle, which length is picked up with the compass and transferred to the new figure?
Answer:
The length BC, the distance between the two points where the arc cuts the arms of the given angle. Copying it makes the triangles congruent by SSS.

Teacher's Note:
1. It is the chord of the arc.
2. Same radius plus same chord gives the same angle.

Question 17. Which pair of angles is made equal while constructing a line parallel to a given line with a ruler and a compass?
Answer:
A pair of corresponding angles made by the transversal with the given line and with the new line.

Teacher's Note:
1. Equal corresponding angles give parallel lines.
2. The transversal is drawn first.

Question 18. While constructing a 90° angle at a point O of a line, why is only one pair of intersecting arcs enough?
Answer:
Because X and Y are marked with OX = OY, O already lies on the perpendicular bisector of XY, so only one more point of that line is needed.

Teacher's Note:
1. Two points fix a line.
2. O is the midpoint of XY.

Question 19. Why must the angles of the tiles that meet at a point in a tiling add up to exactly 360°?
Answer:
The complete angle around a point is 360°. If the angles add to less, a gap is left; if they add to more, tiles overlap.

Teacher's Note:
1. Less than 360° means a gap.
2. More than 360° means an overlap.

Question 20. The front face of a beehive is tiled with regular hexagonal cells. If each angle of a regular hexagon is 120°, how many cells meet at a corner point of the tiling?
Answer:
Three cells, since 3 × 120° = 360°.

Teacher's Note:
1. Divide 360° by the angle.
2. No space in the comb is wasted.

Question 21. Can a grid of 4 rows and 7 columns be tiled with 2 × 1 tiles? Give a reason.
Answer:
Yes. Each column has 4 squares, an even number, so each of the 7 columns takes 2 vertical tiles: 28 ÷ 2 = 14 tiles in all.

Teacher's Note:
1. One even side is enough.
2. Fill along the even side.

Question 22. What supporting lines are needed to construct a pointed arch?
Answer:
Only two line segments of equal length, with their midpoints marked; the arcs drawn on them complete the arch.

Teacher's Note:
1. Changing the radii changes the shape of the arch.
2. Midpoints are found by perpendicular bisectors.

Question 23. Write the two conditions that the supporting lines AB, CD of a trefoil arch must satisfy for the arch to be symmetrical.
Answer:
AB = CD and ∠BAD = ∠CDA: equal lengths and equal angles at A and D.

Teacher's Note:
1. One condition on length, one on angle.
2. The equal angle is made by copying an angle.

Question 24. The 6-pointed star is drawn by first constructing which regular polygon?
Answer:
A regular hexagon. The six points of the star are equilateral triangles built on its six sides.

Teacher's Note:
1. The hexagon gives six equal sides to build on.
2. Each point of the star is a 60° triangle.

Question 25. The unit squares of a region are coloured black and white like a chessboard, and the region has 9 black squares and 7 white squares. Why can this region never be tiled with 2 × 1 tiles?
Answer:
Every 2 × 1 tile covers one black and one white square, so a tiled region needs equal numbers of each. Here 9 ≠ 7, so tiling is impossible.

Teacher's Note:
1. 16 squares is even, yet the tiling fails.
2. Colour counting is stronger than plain counting.

Question 26. Each angle of a regular pentagon is 108°. Give the reason why copies of a regular pentagon alone cannot tile the plane.
Answer:
Tiles fit at a point only if their angles add up to 360°. But 3 × 108° = 324° and 4 × 108° = 432°, so 108° does not divide 360° exactly; pentagons always leave a gap or overlap.

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

Question 27. Can a grid of 11 rows and 13 columns be tiled with 2 × 1 tiles? Justify your answer.
Answer:
No. The grid has 11 × 13 = 143 unit squares, an odd number, but tiles always cover an even number of squares.

Teacher's Note:
1. Both sides odd means an odd total.
2. One square is always left over.

Question 28. While bisecting ∠AOB, a student cuts the two arcs from A and from B with two different radii and joins O to their meeting point C. Will OC bisect the angle? Give a reason.
Answer:
Not in general. With different radii AC ≠ BC, so △OAC and △OBC are no longer congruent by SSS and ∠AOC and ∠BOC need not be equal.

Teacher's Note:
1. Both arcs must use the same radius.
2. SSS needs AC = BC.

Question 29. In the rope method of the Sulba-Sutras, the midpoint of a rope fixed at X and Y is pulled tight above XY to mark A. Why is A a point of the perpendicular bisector of XY?
Answer:
The two halves of the rope are equal, so AX = AY. A point equidistant from X and Y always lies on the perpendicular bisector of XY.

Teacher's Note:
1. The rope acts like equal compass arcs.
2. Equidistant points lie on the bisector.

Question 30. At one point of a tiling, two tiles whose angles are 150° each meet along with one more tile. Can the third tile be an equilateral triangle? Justify by calculation.
Answer:
Yes. 150° + 150° = 300°, so the space left is 360° − 300° = 60°, which is exactly the angle of an equilateral triangle.

Teacher's Note:
1. Find the gap by subtracting from 360°.
2. 150° is the angle of a regular 12-gon.