NCERT Solutions for Class 6 Maths Chapter 08 Playing with Constructions

Page 188

Question. Think: Imagine marking all the points of 4 cm distance from the point P. How would they look?
Answer: When you mark all the points that are 4 cm away from point P, they form a circle. The point P becomes the centre, and 4 cm becomes the radius of that circle.
In simple words: All points at the same distance from one fixed point make a circle.

Exam Tip: Remember that a circle is defined by all points equidistant from a single centre point - this is the foundation of many construction problems.

Page 191 - Figure It Out

Question. 1. What radius should be taken in the compass to get this half circle? What should be the length of AX?
Answer: The diameter of the semi-circle is 4 cm, so we must use 2 cm in the compass to draw this semi-circle. This means the length of AX is 2 cm.
In simple words: If the full width is 4 cm, half of that is 2 cm. That half-distance is what you set in the compass.

Exam Tip: Always remember: radius is half the diameter. If a semi-circle has a 4 cm diameter, set your compass to exactly 2 cm.

Question. 2. Take a central line of a different length and try to draw the wave on it. 3. Try to recreate the figure where the waves are smaller than a half circle (as appearing in the neck of the figure 'A Person'). The challenge here is to get both the waves to be identical. This may be tricky!
Answer: You can draw waves on any line by using semi-circles of different sizes. For smaller waves (like those in the neck), use semi-circles with a smaller radius than the half-circle used before. The key is to keep all the small semi-circles the same size so they look identical and create a neat wave pattern. You can alternate the semi-circles above and below the line to get the wavy effect shown in the figure.
In simple words: Draw small curves of the same size, one after another, going up and down to make a wave. The trick is making all the curves exactly the same.

Exam Tip: Mark equal spaces on your line before drawing each semi-circle - this ensures all curves are identical and gives a neat, professional look.

Page 192

Question. 3. Eyes. How do you draw these eyes with a compass?
Answer: To draw eyes with a compass, follow these steps:
Start by drawing a vertical straight line. Using a compass with a radius that is more than half the line length, position the compass point at one end of the line and draw an arc. Without changing the compass width, place the compass point at the other end of the line and draw another arc so that the two arcs cross. Repeat this process to form a second eye shape. Once you have both eye outlines, place the compass point at the midpoint and adjust the radius to about 1 cm, then draw a small circle inside each eye. This small circle represents the iris or pupil.
In simple words: Draw two curves that meet at the top and bottom to make an eye shape. Then add a small circle in the middle for the pupil.

Exam Tip: The key is using the same compass width for both arcs - if you change it, the eye shape will be uneven.

Question. Make other artwork of your choice with a ruler and a compass.
Answer: Artwork: Circle Flower Pattern.
Steps to create:
Start by drawing a base circle with a convenient radius (say 4 cm). Mark the centre point (O). Without changing the compass width, place the compass point anywhere on the circumference of the circle and draw small arcs that intersect the circumference. Move the compass point to each new intersection and continue drawing more arcs - this way you get 6 intersection points evenly spaced around the circle. With the same radius setting, place the compass point at one of these intersection points and draw a complete circle. Repeat this process for each of the six intersection points, drawing overlapping circles at each location. You will get a flower-like pattern forming inside the main circle. For a three-dimensional look, shade alternate petals. Outline the main shapes using a pen for clarity.
In simple words: Draw a big circle, then put smaller circles around it at equal spots. The overlapping circles make a pretty flower shape.

Exam Tip: This pattern shows symmetry - examiners value neat, equally-spaced construction work; use light pencil lines first before inking.

Page 193

Question. Which of the following is not a name for this square?
(1) PQSR
(2) SPQR
(3) RSPQ
(4) QRSP
Answer: (1) PQSR
In simple words: A square's name must list corners in order (going around continuously). PQSR jumps around and breaks the order, so it is not a proper name.

Exam Tip: Always name shapes by moving around the vertices in sequence - clockwise or counter-clockwise. Skipping or reversing the order makes it invalid.

Question. 1. Draw the rectangle and four squares configuration (shown in Fig.) on a dot paper. What did you do to recreate this figure so that the four squares are placed symmetrically around the rectangle? Discuss with your classmates.
Answer: To place four squares symmetrically around a rectangle, you simply move the squares near the rectangle so that they are positioned evenly on all sides. Each square should be touching or very close to one side of the rectangle, with equal spacing on the left, right, top, and bottom. This creates a balanced, symmetrical design where the rectangle sits in the middle and the four squares surround it in a mirror-image arrangement.
In simple words: Put one square near each side of the rectangle, making sure they are all the same distance away. This makes it look balanced and neat.

Exam Tip: Symmetry means both sides look the same - check that left and right squares are equidistant from the rectangle, and top and bottom squares are also equidistant.

Question. 2. Identify if there are any squares in this collection. Use measurements if needed.
Answer: To identify squares, measure all four sides and check all four angles. A shape is a square only if all sides are equal in length and all angles are right angles (90 degrees). You can use a ruler to measure sides and a set-square or protractor to check angles. Looking at the figure, shapes A, B, and C appear to be squares when rotated, since they have equal sides and right angles. Shape D is not a square because it is smaller and does not match the properties of a square.
In simple words: Measure to check if all four sides are the same length and all corners are 90 degrees. If yes, it is a square.

Exam Tip: A rotated square is still a square - do not be fooled by the tilt. Check side lengths and angles, not appearance.

Question. 3. Draw at least 3 rotated squares and rectangles on a dot grid. Draw them such that their corners are on the dots. Verify if the squares and rectangles that you have drawn satisfy their respective properties.
Answer: When you draw rotated squares and rectangles with corners on dot grid points, verify that: for squares, all four sides have equal length and all angles are 90 degrees; for rectangles, opposite sides are equal, all angles are 90 degrees, and the diagonals bisect each other. Use a ruler to measure each side and confirm these properties hold. The dot grid helps ensure corners are placed accurately, and you can count dots to verify equal spacing. This exercise confirms that rotation does not change the defining properties of these shapes - a rotated square remains a square.
In simple words: Draw tilted squares and rectangles on dots, then measure to check: squares have all equal sides, rectangles have opposite sides equal, all have 90-degree corners.

Exam Tip: Use a ruler and set-square to verify measurements and angles - do not rely on appearance alone, as a tilted figure can deceive the eye.

Page 197 - Construct

Question. 1. Draw a rectangle with sides of length 4 cm and 6 cm. After drawing, check if it satisfies both the rectangle properties.
Answer: A rectangle with sides 4 cm and 6 cm can be drawn as follows: draw a line segment PQ of 6 cm. From P, draw a line perpendicular to PQ and mark a point at 4 cm distance - call it R. From Q, draw another perpendicular line and mark a point at 4 cm distance - call it S. Join RS to complete the rectangle PQSR. After drawing, verify: opposite sides are equal (PQ = SR = 6 cm, PS = QR = 4 cm), and all four angles are right angles (90 degrees each). Both properties are satisfied, confirming it is a valid rectangle.
In simple words: Draw a 6 cm line, add 4 cm perpendiculars at each end, then join the ends. Check that opposite sides match and all corners are 90 degrees.

Exam Tip: Always use a set-square to draw perpendiculars - this ensures right angles, which is the key property of rectangles.

Question. 2. Draw a rectangle of sides 2 cm and 10 cm. After drawing, check if it satisfies both the rectangle properties.
Answer: To construct a rectangle with sides 2 cm and 10 cm: draw a line segment AB of 10 cm length. At point A, draw a perpendicular line and mark a point D at 2 cm distance. At point B, draw another perpendicular and mark a point C at 2 cm distance. Join DC to complete the rectangle ABCD. Verify the properties: opposite sides are equal (AB = DC = 10 cm, AD = BC = 2 cm), and all angles measure 90 degrees. This confirms the properties are satisfied.
In simple words: Draw a 10 cm base, add 2 cm perpendiculars at both ends, then join them. Check opposite sides and right angles.

Exam Tip: For very thin or very wide rectangles, ensure perpendiculars are drawn carefully - a small error becomes noticeable in extreme proportions.

Question. 3. Is it possible to construct a 4-sided figure in which all the angles are equal to 90° but opposite sides are not equal?
Answer: It is not possible to construct such a 4-sided figure. If all angles in a quadrilateral are 90 degrees, the figure must be either a square or a rectangle. In both cases, opposite sides are always equal. This is a defining property: when all four angles are right angles, the opposite sides must be equal. Therefore, you cannot have a 4-sided figure where all angles are 90 degrees without having opposite sides equal.
In simple words: If all four corners are right angles, the shape must be a square or rectangle - and both always have opposite sides equal. You cannot break this rule.

Exam Tip: This is a key geometric principle - always recall that right angles in a quadrilateral force opposite sides to be equal.

Page 198 - An Exploration in Rectangles

Question. At which positions will the points X and Y be at their closest? When do you think they will be the farthest? What does your intuition say? Discuss with your classmates.
Answer: Points X and Y will be closest when XY equals AB, meaning they have both travelled the same distance. They will be farthest when X is at point A and Y is at point C, or when X is at point D and Y is at point B. Your intuition should tell you that the distance between X and Y changes as they move along the sides - it increases and decreases in a pattern. When both have moved equal distances along their respective sides, they are nearest; when one is at a corner and the other is at the opposite end, they are farthest.
In simple words: X and Y are closest when they have moved the same distance. They are farthest when one is at a corner and the other is far away.

Exam Tip: Use the distance formula to verify your intuition - calculate XY for different positions to see the pattern of closest and farthest distances.

Page 199

Question. In each of these cases, observe (i) how the length XY compares to that of AB and (ii) the shape of the 4-sided figure ABYX.
Answer: (i) The length XY is always greater than or equal to AB. (ii) The shape of the 4-sided figure ABYX is a trapezium or a rectangle.
In simple words: XY is never shorter than AB. The shape changes between a trapezium and a rectangle depending on where X and Y are.

Exam Tip: Observe that when X and Y are at corresponding positions (equal distances from their start points), ABYX becomes a rectangle; otherwise it is a trapezium.

Question. How does the farthest distance between X and Y compare with the length of AC? BD?
Answer: The farthest distance between X and Y will be equal to both AC and BD. This is because the diagonals of a rectangle are equal in length, and the maximum separation of X and Y occurs when they are positioned at opposite corners, matching the diagonal length.
In simple words: The farthest X and Y can be apart is as long as the diagonal of the rectangle.

Exam Tip: Remember that in a rectangle, both diagonals have equal length - this property helps determine maximum distances in such explorations.

Page 199 - Breaking Rectangles

Question. Construct a rectangle that can be divided into 3 identical squares as shown in the figure. Solution: If this seem difficult, let us simplify the problem.
Answer: To construct a rectangle that divides into 3 identical squares, mark two points at equal distances on a pair of opposite sides of the rectangle and join them with perpendiculars. If each square has side length s, then the rectangle must have dimensions s by 3s (one side equals the square side, the other side is three times the square side). For example, if the square side is 2 cm, draw a rectangle with sides 2 cm and 6 cm. Mark points P and Q on side AB at distances 2 cm and 4 cm from A, then draw perpendiculars to side DC at corresponding points M and L. These perpendiculars divide the rectangle into three equal squares of 2 cm by 2 cm each.
In simple words: Make a long thin rectangle that is three times as long as it is wide. Then divide it into three equal parts with vertical lines.

Exam Tip: The length of the rectangle must be exactly three times the width - measure carefully to ensure the three resulting sections are truly identical squares.

Question. Explore: What about constructing a rectangle that can be divided into two identical squares? Can you try it? It is wise to first plan and then construct. But how do we plan? Can you think of a way?
Answer: To construct a rectangle divided into two identical squares, the length of the rectangle must be exactly double its breadth. If the square has side length s, the rectangle dimensions are s by 2s. For instance, if you want each square to be 3 cm by 3 cm, construct a rectangle of 3 cm by 6 cm. Draw a line down the middle, connecting the midpoints of the longer sides - this divides it into two equal squares. Planning involves deciding the square size first, then making the rectangle length twice the width before constructing.
In simple words: A rectangle can be split into two equal squares only if it is twice as long as it is wide. Divide it down the middle.

Exam Tip: Always sketch a rough plan first showing the dimensions - this prevents mistakes during the actual construction.

Question. Explore: Can the rectangle now be completed?
Answer: Yes, the rectangle can be completed once you have divided it into two identical squares. By drawing a vertical line through the midpoint of the length, the rectangle ABCD becomes fully divided, with the line showing clearly how the two squares fit together. The rectangle is now complete in its construction and division.
In simple words: Yes, once you draw the middle line, the rectangle is finished and split into two equal squares.

Exam Tip: Ensure the dividing line is perpendicular to the sides and passes through exact midpoints for a clean, correct division.

Page 200

Question. Explore: Can the rectangle now be completed?
Answer: Yes, the rectangle is complete when its length is exactly twice its breadth. At this stage, the rectangle is fully drawn and can be divided into two identical squares by drawing a line through the middle. The construction shows all four sides, all angles are right angles, and opposite sides are equal, making it a valid rectangle.
In simple words: When the length is exactly double the width, the rectangle is done and ready to split into squares.

Exam Tip: Double-check dimensions before completing - measure to confirm length is exactly twice the breadth.

Question. With this idea, try constructing a rectangle that can be divided into three identical squares.
Answer: Using the same principle, a rectangle that divides into three identical squares must have its length exactly three times its breadth. If each square has side s, build a rectangle with dimensions s by 3s. For example, using 2 cm squares, construct a rectangle 2 cm wide and 6 cm long. Divide it into three equal sections by drawing two vertical lines at 2 cm and 4 cm from one end. Each section will be a 2 cm by 2 cm square. This method extends the earlier two-square concept to any number of identical squares.
In simple words: Make a rectangle that is three times as long as it is wide. Divide it with two lines to get three equal squares.

Exam Tip: The dividing lines must be equally spaced and perpendicular to the longer sides - precision ensures all resulting squares are identical.

Page 201

Question. Construct - A Square within a Rectangle. Construct a rectangle of sides 8 cm and 4 cm. How will you construct a square inside, as shown in the figure, such that the centre of the square is the same as the centre of the rectangle? Hint: Draw a rough figure. What will be the side length of the square? What will be the distance between the corners of the square and the outer rectangle?
Answer: To construct a square inside the rectangle with the same centre: first draw the rectangle with sides 8 cm and 4 cm. The centre of the rectangle is at the midpoint of both diagonals. The square that shares this centre will have a side length of 4 cm (equal to the shorter side of the rectangle). Mark arcs of 2 cm radius from each corner of the rectangle, centred at each corner. These arcs intersect the longer sides at four points. Connecting these four intersection points creates the required square. The distance between the corners of the square and the outer rectangle is 2 cm (measured along the longer sides).
In simple words: Draw an 8 cm by 4 cm rectangle. Mark 2 cm from each corner along the long sides. Connect these four marks to make a square inside with the same centre.

Exam Tip: Use compasses to mark equal arcs from each corner - this ensures the square is centred correctly and all sides are equal.

Question. Give the lengths of the sides of a rectangle that cannot be divided into - (i) two identical squares; (ii) three identical squares.
Answer: A rectangle cannot be divided into two identical squares if its length is not exactly double its width. For example, a rectangle with sides 5 cm and 3 cm cannot be divided into two identical squares. A rectangle cannot be divided into three identical squares if its length is not exactly three times its width. For example, a rectangle with sides 7 cm and 2 cm cannot be divided into three identical squares. In general, the ratio of length to width must equal the number of squares required for a successful division.
In simple words: If length is not double the width, you cannot split it into two equal squares. If length is not triple the width, you cannot split it into three equal squares.

Exam Tip: Always check the ratio of sides before attempting division - a quick calculation saves time and prevents construction errors.

Page 204 - Explore

Question. How should the rectangle be constructed so that the diagonal divides the opposite angles into equal parts?
Answer: For a diagonal to divide the opposite angles into equal parts, the rectangle must actually be a square. In a square, the diagonals bisect all angles - each diagonal divides the corner angles into two equal 45-degree angles. If the diagonal divides the opposite angles into equal parts, the angles at those corners must be equal, which forces all angles to be equal (90 degrees each). Combined with equal sides, this means it must be a square.
In simple words: If a diagonal cuts opposite corners into equal angles, the shape is a square, not just a rectangle.

Exam Tip: Recall that only in a square do diagonals bisect the corner angles - this property distinguishes squares from other rectangles.

Question. What general laws did you observe with respect to the angles and sides? Try to frame and discuss them with your classmates. How can one be sure if the laws that you have observed will always be true?
Answer: By constructing and exploring various rectangles, you observe that as the longer side of a quadrilateral decreases (approaching the length of the shorter side), the angles formed by the diagonal change. Specifically, the difference between the two angles created by a diagonal diminishes. When the longer side equals the shorter side (becoming a square), the two angles are equal. These laws hold because they follow from the geometric properties of similar triangles and angle relationships in polygons. To verify these observations always hold true, use deductive reasoning based on properties of parallel lines, angle bisectors, and the definition of rectangles and squares.
In simple words: As a rectangle gets more and more square-shaped, the diagonal cuts the angles into more equal parts. This is always true because of how angles work in geometry.

Exam Tip: Support your geometric observations with angle measurements and calculations - this transforms intuition into mathematical proof.

Page 211 - Construct

Question. 1. Construct a rectangle in which one of the diagonals divides the opposite angles into 50° and 40°.
Answer: To construct this rectangle: draw a line segment AB of arbitrary length. At point A, draw an angle of 90 degrees and at point B also draw an angle of 90 degrees (using a set-square). From A, draw a line perpendicular to AB. From B, draw another line perpendicular to AB. On the line through A, mark a point D. On the line through B, mark a point C such that ABCD forms a rectangle. Now, to ensure one diagonal divides the opposite angles into 50 degrees and 40 degrees, position D and C such that the diagonal AC creates these angle measures at the corners it connects. Use a protractor to verify the angles. Join D and C to complete the rectangle.
In simple words: Draw a rectangle with perpendicular sides. Then draw a diagonal and use a protractor to check that it splits the opposite corners into 50 degrees and 40 degrees.

Exam Tip: Use a protractor to measure angles precisely - small errors in angle measurement will distort the rectangle's shape.

Question. 2. Construct a rectangle in which one of the diagonals divides the opposite angles into 45° and 45°. What do you observe about the sides?
Answer: To construct a rectangle where a diagonal divides opposite angles into 45 degrees and 45 degrees each: draw a line segment AB of arbitrary length. Draw perpendiculars at A and B, both 90 degrees. On these perpendiculars, mark points D and C such that AD equals AB (making all sides equal). Complete the rectangle by joining all sides. When you draw the diagonal AC, it will divide the opposite angles into 45 degrees and 45 degrees. What you observe is that the four sides are now all equal in length - this is a square. A diagonal in a square always bisects the corner angles equally.
In simple words: When a diagonal splits opposite corners into 45 and 45 degrees, all four sides turn out to be equal - it is a square.

Exam Tip: This construction confirms the key property: equal angle bisection by a diagonal only occurs in a square, never in a non-square rectangle.

Question. 3. Construct a rectangle one of whose sides is 4 cm and the diagonal is of length 8 cm. After drawing, check if it satisfies both the rectangle properties.
Answer: To construct a rectangle with one side 4 cm and diagonal 8 cm: draw a line segment AB of 4 cm. Draw perpendiculars at both A and B. Using the Pythagorean theorem, calculate the other side: if one side is 4 cm and the diagonal is 8 cm, then the other side is \( \sqrt{8^2 - 4^2} = \sqrt{64 - 16} = \sqrt{48} \approx 6.93 \) cm. Mark this length on the perpendiculars at A and B to get points D and C. Join DC to complete the rectangle ABCD. Verify: opposite sides are equal (AB = DC = 4 cm, AD = BC ≈ 6.93 cm), all angles are 90 degrees, and the diagonal length is 8 cm. Both rectangle properties are satisfied.
In simple words: Draw a 4 cm base. Use the diagonal length of 8 cm and math to find the height. Add perpendiculars of that height, then join the top points.

Exam Tip: Apply the Pythagorean theorem - knowing one side and the diagonal, you can always find the other side of a rectangle.

Question. 4. Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.
Answer: To construct a rectangle with one side 3 cm and diagonal 7 cm: draw a line segment AB of 3 cm. Draw right angles at A and B. Using the Pythagorean theorem: if one side is 3 cm and the diagonal is 7 cm, the other side is \( \sqrt{7^2 - 3^2} = \sqrt{49 - 9} = \sqrt{40} \approx 6.32 \) cm. Mark this length on the perpendiculars from A and B to get points D and C respectively. Join DC to complete rectangle ABCD. The construction has one side 3 cm, the other side approximately 6.32 cm, and the diagonal measures exactly 7 cm.
In simple words: Draw a 3 cm line, add perpendiculars of about 6.32 cm (found using the diagonal and math), then connect the top points.

Exam Tip: For non-standard diagonal lengths, calculate the second side using the Pythagorean theorem before constructing - this ensures accuracy.

Page 213 - Think

Question. Was it necessary to draw two full circles to get the point A? We only needed part of both the circles.
Answer: No, it was not necessary to draw two full circles to find point A. You only needed to draw an arc (a part of the circle) from each centre. Once the two arcs intersect at point A, that intersection point is found, and you do not need to continue drawing the remaining portions of the circles. Drawing only the necessary arcs saves time and keeps the construction neat and clean, with less confusion from extra lines.
In simple words: You only need small curve pieces from each circle to find where they meet - there is no need to draw whole circles.

Exam Tip: In geometric constructions, draw only what is necessary - partial arcs are sufficient to find intersections, and this improves clarity and reduces errors.

Page 214 - Construct

Question. 1. Construct a bigger house in which all the sides are of length 7 cm.
Answer: To build a bigger house where all sides measure 7 cm: start by drawing the rectangular base DEEF with width 7 cm and a suitable height (say 3 cm). Draw a line segment DE of 7 cm. At D and E, draw perpendiculars of 3 cm upward, marking points D and E at the top. Join these to complete the rectangle. For the triangular roof, at the midpoint of the top line, draw a vertical line segment upward. Using a compass set to 7 cm radius, place the point at D and draw an arc above. Place the compass point at E and draw another arc - where the two arcs meet is the apex A of the roof. Join A to D and A to E. All resulting sides (base 7 cm, perpendiculars, and roof sides AD and AE) measure 7 cm, creating the house shape.
In simple words: Draw a 7 cm wide rectangle for the base. Then add a triangular roof by finding the point where two 7 cm arcs from the top corners meet.

Exam Tip: Use compass arcs to ensure the roof sides equal 7 cm - this construction combines rectangle and circle arc techniques.

Question. 2. Try to recreate 'A Person', 'Wavy Wave' and 'Eyes' from the section Artwork, using ideas involved in the 'House' construction.
Answer: To recreate 'A Person' using house construction ideas: the head can be a circle (or semi-circles meeting at a point like the house roof), the body is a rectangle, the arms and legs are line segments or thin rectangles. For 'Wavy Wave', use the semi-circle wave technique from earlier - draw repeated semi-circles along a baseline, alternating above and below, using equal spacing. For 'Eyes', use two curved shapes (like the almond eyes drawn earlier with intersecting circles) positioned symmetrically on the head. Combine these elements: place the eyes in the circular head, add a wavy neck region using semi-circles, attach a rectangular body below, and add simple leg segments. All ideas (circles, rectangles, semi-circles, arcs, and symmetry) from the house construction carry over to these figures.
In simple words: Use circles for the head, rectangles for the body, waves for the neck, and eyes made from curves - just like the house but put together as a person.

Exam Tip: Combine different construction techniques learnt throughout the chapter - circles, rectangles, arcs, and perpendiculars - to create complex artwork.

Question. 3. Is there a 4-sided figure in which all the sides are equal in length but is not a square? If such a figure exists, can you construct it?
Answer: Yes, such a 4-sided figure exists and is called a rhombus. A rhombus has all four sides equal in length, but the angles are not right angles (except in the special case of a square). To construct a rhombus: draw a line segment of any length (say 4 cm). At one end, draw a line at any angle other than 90 degrees (say 60 degrees). On this line, mark a point 4 cm away. From this point, draw a line parallel to the first segment, also of length 4 cm. Join the end of this line back to the starting point to complete the rhombus. All four sides equal 4 cm, but the angles are not 90 degrees (they are 60 degrees and 120 degrees alternately), so it is not a square.
In simple words: A rhombus has all equal sides but tilted corners (not 90 degrees). It looks like a diamond shape.

Exam Tip: Always distinguish between a square (equal sides AND 90-degree angles) and a rhombus (equal sides only, angles tilted).