CBSE Class 6 Mathematics Playing with Constructions Assignment Set 01

Short Answer Questions

Question 1. Rani has to draw a circle of radius 4 cm with centre O in her notebook.
(i) Write the steps she should follow with a ruler and a compass.
(ii) After the curve is drawn, how can she use the ruler to check that it is correct?
Answer:
(i) Mark the point O with a sharp pencil.
Open the compass against the ruler's scale so that the gap between the metal tip and the pencil tip is exactly 4 cm.
Press the metal tip firmly at O and turn only the pencil arm all the way round. The curve drawn is the required circle.
(ii) Pick a few points here and there on the curve and measure each one's distance from O with the ruler. Each must be 4 cm, because every point of a circle is the same distance from its centre.
In simple words: Set the compass to 4 cm, keep the tip at O and turn; then check a few points are 4 cm from O.

Teacher's Note:
1. Do not let the tip slip while turning.
2. Checking several points tests the whole circle.

Question 2. Explain, with the help of the properties R1, R2, S1 and S2, why every square may also be called a rectangle, while a rectangle cannot always be called a square. Give one example of a rectangle that is not a square.
Answer:
A rectangle needs R1 (opposite sides equal) and R2 (all angles 90°). A square needs S1 (all four sides equal) and S2 (all angles 90°).
In a square all sides are equal, so the opposite sides are surely equal (R1), and all angles are 90° (R2). So every square is a rectangle.
A rectangle may have neighbouring sides of different lengths, and then S1 fails. For example, a rectangle of sides 9 cm and 4 cm is not a square.
In simple words: A square passes both rectangle tests, but a rectangle with unequal sides fails the square test S1.

Teacher's Note:
1. Match each property one by one.
2. Any rectangle with unequal neighbouring sides is a valid example.

Question 3. In a rectangle PQRS, PQ = 12 cm and QR = 7.5 cm. Sahil says that if the figure is drawn very carefully it will come out to be a square.
(i) Write the lengths of RS and SP.
(ii) Explain why PQRS can never be a square.
Answer:
(i) RS = PQ = 12 cm and SP = QR = 7.5 cm, because opposite sides of a rectangle are equal (R1).
(ii) A square needs all four sides equal (S1). Here two sides are 12 cm and two are 7.5 cm, and careful drawing cannot change the given lengths. So PQRS can never be a square.
In simple words: The sides are 12, 7.5, 12, 7.5 cm, which are not all equal, so it cannot be a square.

Teacher's Note:
1. Care in drawing improves accuracy, not the lengths given.
2. S1 is the property that fails.

Question 4. The artwork ‘A Person’ is made of two parts: a circle for the head, and below it a 4-sided body whose top line is replaced by a curve that dips inwards.
(i) Which instrument draws the head and which instrument draws the straight sides of the body?
(ii) The curve has to pass neatly through the two upper corners of the body. Write how you will find the place for the tip of the compass.
Answer:
(i) The head is a circle, so it is drawn with the compass. The straight sides of the body are drawn with the ruler, with a set square or protractor to make the corners 90°.
(ii) The curve must pass through both upper corners and dip inwards, so the compass tip must lie somewhere above the body.
Choose a radius, place the tip at an estimated point and draw. If the arc does not pass through both corners, shift the tip a little or change the radius and try again, until the curve dips as required.
In simple words: Compass for the head, ruler for the body; find the curve's centre above the body by careful trying.

Teacher's Note:
1. The tip should be at equal distances from the two corners.
2. Draw trial arcs lightly.

Question 5. A compass is used again and again in this chapter.
(i) Write two jobs that a compass can do here which a ruler cannot do.
(ii) Write one job for which the compass and the ruler have to be used together.
Answer:
(i) It draws circles and arcs of a chosen radius, which a ruler cannot draw at all.
It carries a length from one part of a figure to another without measuring, for example from one side of a square to the next.
(ii) To set the compass to an exact radius, such as 4 cm, it is opened against the scale of the ruler, so both are used together.
In simple words: A compass draws curves and copies lengths; it needs a ruler to be set to an exact size.

Teacher's Note:
1. Drawing curves and copying lengths are the two main jobs.
2. The ruler gives the exact number.

Question 6. On dot paper Meera draws a 4-sided figure whose corners are all on dots, and she says it is a square. Explain how one can decide whether its sides are equal and its corners are right angles by looking only at the positions of the corners on the dot grid, without using any measuring instrument.
Answer:
Yes, the dots alone are enough.
Count the dot steps from one corner to the next, for example 3 steps right and 1 step up.
If every side needs the same pair of steps, all four sides are equal.
If at each corner the pair of steps swaps and turns, such as ‘3 right, 1 up’ followed by ‘1 left, 3 up’, the figure makes a quarter turn there, so that corner is a right angle.
When this happens at all four corners, the figure is a square, without any ruler or protractor.
In simple words: Same step-pattern on every side means equal sides; swapped-and-turned steps at each corner mean right angles.

Teacher's Note:
1. Write the steps for each side as a pair of numbers.
2. A tilted square on dot paper is still a square.

Question 7. In the ‘Wavy Wave’ figure the central line is AB with AB = 8 cm. The first wave is a half circle drawn above the line from A to the point X, where AX = 4 cm, and the second wave is a half circle drawn below the line from X to B. The compass is kept open at 2 cm all through.
(i) Where should the tip of the compass be placed for each of the two waves?
(ii) Why does the first wave come back to the central line exactly at X?
Answer:
(i) First wave: at the mid-point of AX, that is 2 cm from A on the central line.
Second wave: at the mid-point of XB, that is 6 cm from A (or 2 cm from B).
(ii) That tip is 2 cm from A and also 2 cm from X, and the radius is 2 cm. So both A and X lie on the circle, and the half circle starting at A meets the line again exactly at X.
In simple words: Put the tip at the middle of each 4 cm piece; with radius 2 cm the curve hits both ends.

Teacher's Note:
1. The centre of a half circle is the mid-point of its diameter.
2. Keep the radius at 2 cm for both waves.

Question 8. The diagonals PR and QS of a rectangle PQRS are drawn, and PR measures 10 cm.
(i) What is the length of QS? Give a reason.
(ii) A square of side 6 cm is drawn and both its diagonals are drawn. What can you say about their lengths? Give a reason.
Answer:
(i) QS = 10 cm, because the two diagonals of a rectangle are always equal in length.
(ii) The two diagonals of the square are also equal in length. A square has opposite sides equal and all angles 90°, so it is also a rectangle, and the diagonals of a rectangle are equal.
In simple words: Diagonals of any rectangle, including a square, are equal.

Teacher's Note:
1. A square is a special rectangle, so it shares this property.
2. Measure to confirm in your notebook.

Question 9. A rectangle of length 12 cm is to be divided into three identical squares.
(i) Find the breadth of the rectangle and the side of each square.
(ii) Explain how a rough diagram helps you to decide these lengths before drawing.
Answer:
(i) The three squares stand side by side along the length, so the length is 3 times the side of a square.
Side of each square = 12 ÷ 3 = 4 cm, and the breadth of the rectangle = 4 cm.
(ii) In the rough diagram the rectangle is cut by two lines into three equal squares. It shows at once that all the short lines are equal and that the long side is made of three of them, so we know the lengths before we start.
In simple words: 12 ÷ 3 = 4 cm is both the square's side and the breadth; the rough sketch shows this.

Teacher's Note:
1. Mark equal sides with small strokes.
2. Length = 3 × breadth for three squares.

Question 10. A design starts with a big square of side 8 cm which is divided into four identical smaller squares.
(i) What is the side of each smaller square?
(ii) Write how you will draw the two lines that divide the big square, naming the instruments used.
Answer:
(i) Two small squares stand along each side, so each side is 8 ÷ 2 = 4 cm.
(ii) With the ruler, mark the mid-points of the top and bottom sides (4 cm from a corner) and join them.
In the same way mark the mid-points of the left and right sides and join them with the ruler.
A set square may be used to check that each line is perpendicular to the sides it meets. The big square is now four squares of side 4 cm.
In simple words: Join the mid-points of opposite sides to get four 4 cm squares.

Teacher's Note:
1. The two lines cross at the centre of the square.
2. Check each line meets the sides at 90°.

Question 11. In the ‘Falling Squares’ design, three squares of side 4 cm are drawn in a staircase way, so that a corner of one square touches a corner of the next one. The lowest square has already been drawn. Write how you will draw the second square in its correct position, naming the instruments used.
Answer:
Take the top right corner of the first square as the bottom left corner of the second square.
From this corner draw a line of 4 cm towards the right with the ruler.
At both ends of this line draw perpendiculars going upwards with the set square.
Cut off 4 cm on each perpendicular with the compass opened to 4 cm, and join the two new points with the ruler to close the square.
In simple words: Start the new square at the old square's top right corner and build a 4 cm square from there.

Teacher's Note:
1. The corner shared by the two squares is the key point.
2. Use the same steps as for any square.

Question 12. The two curves of an ‘eye’ in the Artwork section are drawn with a compass. The upper curve bulges upwards and the lower curve bulges downwards, and both curves pass through the two corner points of the eye.
(i) Where should the tip of the compass be placed for these two curves?
(ii) What must be kept the same so that the eye looks symmetrical?
Answer:
(i) Each curve passes through both corner points, so its centre must be at the same distance from these two points.
So the tip goes on the line drawn at right angles through the middle of the join of the corners: below the eye for the upper curve and above the eye for the lower curve.
(ii) The two tips must be at equal distances from the eye and the same radius must be used for both arcs. Then the two curves match.
In simple words: Put the centre on the middle line, below for the top curve and above for the bottom one, using the same radius.

Teacher's Note:
1. An arc bulges away from its centre.
2. Same radius and same distance give a matching pair.

Question 13. Pooja is constructing a square PQRS of side 5 cm. She has drawn PQ, then the perpendiculars at P and at Q, and has marked S and R on them. Only the last side SR is left to be drawn.
(i) Without measuring, write the length of SR and the measures of ∠R and ∠S.
(ii) Give a reason for your answer.
Answer:
(i) SR = 5 cm, ∠R = 90° and ∠S = 90°.
(ii) PS and QR are equal and both stand at right angles on PQ, so S and R are at the same height above PQ.
So SR runs level with PQ and is equal to it, 5 cm. Being level, it meets the two perpendiculars at right angles, so ∠R = ∠S = 90°. The figure satisfies S1 and S2.
In simple words: S and R are at the same height, so SR is level, 5 cm long, and makes right angles.

Teacher's Note:
1. Equal perpendiculars give a level top side.
2. Still check by measuring after drawing.

Question 14. Two points B and C are marked 4 cm apart. An arc of radius 3 cm is drawn with the compass tip at B, and another arc of radius 3 cm with the tip at C.
(i) How many points lie on both the arcs, and where do they lie?
(ii) What is special about each of these points?
Answer:
(i) Two points: one above the line BC and one below it.
(ii) Each point is 3 cm from B and also 3 cm from C, because every point of the first arc is 3 cm from B and every point of the second is 3 cm from C.
The arcs do meet, since 3 + 3 = 6 cm is more than the 4 cm gap between B and C.
In simple words: The arcs cross twice, once on each side of BC, at points 3 cm from both B and C.

Teacher's Note:
1. Two arcs of equal radius cross on both sides of the line.
2. They meet only if the two radii add up to more than the gap.

Question 15. Ravi has drawn a 4-sided figure and says that it is a square. Describe the checks he should make before he can be sure, naming the instrument used for each check.
Answer:
Measure all four sides with a ruler; all four must be equal (S1).
Measure all four angles with a protractor, or test each corner with a set square; each must be 90° (S2).
A compass is handy to compare sides: open it to one side and carry that length to the other three.
Only when both S1 and S2 hold can Ravi call his figure a square.
In simple words: Check four equal sides and four right angles before calling it a square.

Teacher's Note:
1. Checking only the sides is not enough.
2. Name an instrument for each check.

Question 16. A diagonal of a rectangle divides one of its right angles into two parts measuring 32° and 58°.
(i) Into which two parts does the same diagonal divide the right angle at the opposite corner?
(ii) Give a reason for your answer.
Answer:
(i) Into the same two parts, 32° and 58°.
(ii) The two parts of a right angle cut by a diagonal must add up to 90°, and 32° + 58° = 90°.
On measuring we find that the same pair of parts appears again at the opposite corner, with their places interchanged: if the 32° part at the first corner lies next to one long side, the 32° part at the opposite corner lies next to the other long side (the side opposite to it).
In simple words: The opposite corner is split into the same 32° and 58°, placed the other way round.

Teacher's Note:
1. The parts always add up to 90°.
2. Draw a rectangle and its diagonal, and measure to see the swap.

Question 17. In a rectangle ABCD, AB = 10 cm and BC = 6 cm. The point X lies on AD at a distance of 1 cm from A, and the point Y lies on BC at a distance of 4 cm from B. Without measuring, decide whether XY is equal to, less than or more than 10 cm, and explain your thinking.
Answer:
XY is more than 10 cm.
X is 1 cm from A and Y is 4 cm from B, so they are not at the same level; Y is 3 cm farther along than X.
When the two distances are equal, XY runs straight across and equals AB = 10 cm, the shortest it can ever be.
Here the line also slopes by 3 cm, so it is a slanting line and must be longer than 10 cm.
In simple words: Unequal distances make XY slant, so it is longer than the straight-across 10 cm.

Teacher's Note:
1. Straight across is the shortest.
2. Any slant makes the line longer.

Question 18. A rectangle is to be constructed with one side 4.5 cm and a diagonal of given length.
(i) In any rectangle, which is longer, a side or a diagonal? Give a reason.
(ii) Can the rectangle be drawn if the diagonal is only 3 cm long? Explain what happens to the arc in that case.
Answer:
(i) The diagonal is longer. A side joins two neighbouring corners, while a diagonal goes right across to the opposite corner. On drawing and measuring it always comes out longer; for example, with sides 6 cm and 8 cm the diagonal is 10 cm.
(ii) No. After drawing the 4.5 cm side and the perpendicular at one end, an arc of radius 3 cm from the other end is too short to reach the perpendicular.
The arc and the line never meet, so the corner cannot be marked.
In simple words: A diagonal is always longer than a side, so a 3 cm diagonal with a 4.5 cm side is impossible.

Teacher's Note:
1. The perpendicular is 4.5 cm away from the arc's centre at its nearest point.
2. Check that diagonal > side before starting.

Question 19. Three points are marked so that AB = 5 cm, AC = 5 cm and ∠BAC is clearly not a right angle. A fourth point D has to be marked so that ABDC becomes a 4-sided figure with every side 5 cm.
(i) Write how the point D is found with a compass.
(ii) Will ABDC be a square? Give a reason.
Answer:
(i) Open the compass to 5 cm, the length of AB.
With the tip at B, draw an arc on the side away from A; with the same opening and the tip at C, draw another arc.
The crossing point is D, as it is 5 cm from both B and C. Join BD and CD.
(ii) No. All four sides are 5 cm, so S1 holds, but ∠BAC is not 90°, so S2 fails. Such a slanting figure with four equal sides is called a rhombus.
In simple words: Two 5 cm arcs from B and C fix D; the figure is a rhombus, not a square, as its angles are not 90°.

Teacher's Note:
1. Draw the arcs on the side away from A.
2. Equal sides alone do not make a square.

Question 20. In the ‘Square with Curves’ figure, a square of side 8 cm has one arc drawn on each of its four sides, and every arc passes through the two corners of that side and bulges into the square.
(i) On which line must the tip of the compass be placed to draw the arc of one side?
(ii) What must be kept the same for all the four arcs, and why?
Answer:
(i) The arc passes through both corners of a side, so its centre must be the same distance from those two corners.
So the tip must be on the line through the middle of that side, at right angles to it, and outside the square so that the arc bends inwards.
(ii) The same radius must be used for all four arcs, and every tip must be at the same distance from its own side (that is, from the centre of the square). Then all four arcs bulge in by the same amount and the figure looks alike from every side.
In simple words: Put each centre on the side's perpendicular middle line, outside the square, with the same radius every time.

Teacher's Note:
1. An arc bends away from its centre, so the centre is outside.
2. Equal radius and equal distance give four matching arcs.