✅ Answer: Yes — figures are neater, equal and faster to draw.
๐ Textbook Page 190
๐ Page 190Construct · A Person
How will you draw ‘A Person’ (a head, a neck and a body with curved shoulders)?
A Person: head = circle; the curved shoulders = an arc whose centre is above the body.
1Head: a circle; neck: a short straight line down from it.
2Body: draw the two sides and the bottom of a square/rectangle with a ruler.
3Shoulders: the top is an arc. Keep the radius a little more than half the width and place the compass tip above the body on the middle line.
4Adjust the tip up or down until the arc passes through both top corners.
✅ Answer: Circle + line + rectangle, with an arc whose centre lies above the middle of the body.
๐ก Tip: Mark the centre on the middle line so the curve is the same on both sides.
๐ Textbook Page 191
๐ Page 191Wavy Wave · Q1
AB = 8 cm. The first wave is a half circle on AX. What radius should be taken? What is AX?
Wavy wave on AB = 8 cm.
1The two waves are equal, so X is the middle of AB: AX = 8 ÷ 2 = 4 cm.
2The half circle sits on AX, so AX is its diameter.
3Radius = 4 ÷ 2 = 2 cm; centre = middle of AX (2 cm from A).
AB = 8 cm: two half circles of radius 2 cm on AX and XB.
✅ Answer: Radius 2 cm; AX = 4 cm.
๐ Page 191Wavy Wave · Q2
Take a central line of a different length and draw the wave.
1Example: AB = 12 cm → AX = 6 cm, radius 3 cm.
2Centres at 3 cm and 9 cm from A; draw one half circle above and one below.
AB = 12 cm: two half circles of radius 3 cm on AX and XB.
✅ Answer: e.g. AB = 12 cm: radius 3 cm, centres 3 cm and 9 cm from A.
๐ Page 191Wavy Wave · Q3
Make the waves smaller than a half circle (like the neck of ‘A Person’), with both waves identical.
1Use a radius bigger than half of AX, e.g. AX = 4 cm, radius 3 cm.
2Upper wave: centre BELOW the middle of AX. Lower wave: centre ABOVE the middle of XB.
3Keep the same radius and the same distance of the centre from AB for both waves.
✅ Answer: Same radius, centres on opposite sides at the same distance from AB.
๐ Textbook Page 192
๐ Page 192Construct · Eyes
How do you draw these eyes with a compass?
Eyes: each eye is made of two arcs; the compass tip is placed above (for the lower curve) and below (for the upper curve).
1Draw a faint helping line through the eye corners.
2Lower curve: compass tip at a point A above the line, radius reaching both corners.
3Upper curve: tip at point B the same distance below the line, same radius.
4Shade a small circle in the middle. Repeat for the second eye.
✅ Answer: Two equal arcs with centres the same distance above and below the line.
๐ Page 192Squares and rectangles
What makes a figure a rectangle or a square?
Rectangle ABCD
1Rectangle: (R1) opposite sides equal, (R2) all angles 90°.
2Square: (S1) all sides equal, (S2) all angles 90°.
✅ Answer: See the properties R1, R2 and S1, S2.
๐ Textbook Page 193
๐ Page 193Naming a square
Which is NOT a name for the square (corners S, P, Q, R going round)? PQSR, SPQR, RSPQ, QRSP
Square with corners S (top left), P, Q, R.
1A name must go round the square in order (either direction).
2SPQR, RSPQ, QRSP go round in order.
3PQSR jumps from Q to S across the diagonal.
✅ Answer: PQSR
๐ Page 193Rotated squares
A square paper is rotated. Is it still a square?
1Rotating does not change lengths or angles.
2All sides are still equal and all angles still 90°.
✅ Answer: Yes — a rotated square is a square, and a rotated rectangle is a rectangle.
๐ Textbook Page 194
๐ Page 194Figure it Out · Q1
Draw the rectangle with four squares around it (Fig. 8.3) on dot paper, placed symmetrically.
1Draw the rectangle first, e.g. 6 dots by 3 dots.
2Place equal squares (e.g. 2 by 2) so that each one is the same number of dots away from its corner of the rectangle.
3Count dots to check that left and right, top and bottom match.
✅ Answer: Count equal gaps of dots from each corner.
๐ Page 194Figure it Out · Q2
Identify the squares among A, B, C and D. Can we decide just by looking at the dots?
Shapes A–D on dot paper.
1A: each side goes 4 across and 4 up/down — all sides equal, and the turns are right angles → square.
2B: sides go 3 across and 4 down — sides equal, but the diagonals are 6 and 8 dots long, so angles are not 90° → not a square.
3C: 5 across and 4 down — again a diamond with unequal diagonals → not a square.
4D: sides 2 across 1 down and 2 across 4 down — right angles but sides unequal → a rectangle, not a square.
Only A is a square: sides go 4 across and 4 down, all equal, corners 90°.
✅ Answer: Only A is a square (D is a rectangle).
๐ก Tip: On dot paper, a side going ‘a across, b down’ is perpendicular to one going ‘b across, a up’.
๐ Page 194Figure it Out · Q3
Draw at least 3 rotated squares and rectangles on a dot grid with corners on dots and verify them.
1Square: sides 2 across 1 up, then 1 across 2 down, then 2 back 1 down, then 1 back 2 up.
2Rectangle: sides 2 across 2 up and 1 across 1 down (lengths differ).
3Check with a ruler and protractor.
✅ Answer: Many answers — use ‘a across b up’ and ‘b across a down’ for the turns.
๐ Textbook Page 195
๐ Page 195Constructing a square
Construct a square PQRS of side 6 cm.
1Draw PQ = 6 cm.
2Draw a perpendicular at P (protractor: 90°).
3Mark S on it with PS = 6 cm (ruler or compass opened to 6 cm).
4Draw a perpendicular at Q and mark R with QR = 6 cm.
5Join RS.
Square PQRS of side 6 cm: PQ first, perpendiculars at P and Q, then mark 6 cm on each with the compass.
✅ Answer: See the construction.
๐ Textbook Page 196
๐ Page 196Check the square
How long is RS and what are ∠R and ∠S?
1Measure: RS = 6 cm.
2∠R = 90° and ∠S = 90°.
✅ Answer: RS = 6 cm, ∠R = ∠S = 90°
๐ Textbook Page 197
๐ Page 197Construct · Q1
Draw a rectangle with sides 4 cm and 6 cm. Check both properties.
1Draw AB = 6 cm; perpendiculars at A and B; mark 4 cm on each; join.
2Check: opposite sides 6, 6 and 4, 4; all four angles 90°.
Rectangle 6 cm × 4 cm
✅ Answer: See the figure.
๐ Page 197Construct · Q2
Draw a rectangle of sides 2 cm and 10 cm and check its properties.
1Same steps with AB = 10 cm and the perpendiculars 2 cm.
Rectangle 10 cm × 2 cm
✅ Answer: See the figure.
๐ Page 197Construct · Q3
Is it possible to make a 4-sided figure with all angles 90° but opposite sides not equal?
1Draw the bottom side and two perpendiculars at its ends.
2The top side must also be perpendicular to both, so it is parallel to the bottom.
3Then the two perpendicular sides are cut to the same length, and the top equals the bottom.
✅ Answer: No — if all angles are 90°, opposite sides are always equal.
๐ Textbook Page 198
๐ Page 198Exploring X and Y
ABCD has AB = 7 cm, BC = 4 cm; X moves on AD and Y on BC. When are X and Y closest? Farthest? Find XY for the given positions.
X is 0.5 cm below A, Y is 3 cm below B: XY ≈ 7.4 cm.
1XY is shortest when it is straight across, parallel to AB: XY = 7 cm (X and Y at the same distance from A and B).
2XY is longest when X and Y are at opposite corners: X = A, Y = C or X = D, Y = B — the diagonal.
3Diagonal = about 8 cm 1 mm (measure it).
4X 5 mm, Y 3 cm → XY ≈ 7 cm 4 mm; X 1 cm, Y 1 cm → XY = 7 cm 0 mm; X 2 cm, Y 4 cm → XY ≈ 7 cm 3 mm.
Distance of X from A
Distance of Y from B
Length of XY (measured)
0.5 cm
3 cm
7 cm 4 mm
1 cm
1 cm
7 cm 0 mm
2 cm
4 cm
7 cm 3 mm
✅ Answer: Closest: 7 cm (= AB). Farthest: the diagonal ≈ 8 cm 1 mm.
๐ก Tip: Measured answers may differ by 1 mm — that is fine.
๐ Textbook Page 199
๐ Page 199Same distances
When X and Y are the same distance from A and B (5 mm, 1 cm, 1 cm 5 mm), how does XY compare with AB? What shape is ABYX?
1In each case XY = 7 cm = AB.
2ABYX has opposite sides equal and all angles 90°.
X from A
Y from B
XY
5 mm
5 mm
7 cm
1 cm
1 cm
7 cm
1 cm 5 mm
1 cm 5 mm
7 cm
✅ Answer: XY = AB = 7 cm every time; ABYX is a rectangle.
๐ Page 199Farthest distance
How does the farthest distance between X and Y compare with AC and BD?
1The farthest positions are the corners A, C or D, B.
✅ Answer: It is equal to AC = BD (the diagonals, ≈ 8.1 cm).
๐ Textbook Page 200
๐ Page 200Breaking rectangles · 2 squares
Construct a rectangle that can be divided into two identical squares. If AF = 4 cm, what must AC be?
1Rough figure: ACDF with B, E in the middle; all short lines are equal.
2AB = BC = AF, so AC = 2 × AF = 8 cm.
3Draw AF, a perpendicular at A, and with the compass opened to AF mark B and then C.
4Complete the rectangle and join BE.
A rectangle made of 2 equal squares: long side = 2 × short side (equal sides marked |).
✅ Answer: AC = 8 cm (long side = 2 × short side).
๐ Textbook Page 201
๐ Page 201Breaking rectangles · 3 squares
Construct a rectangle that can be divided into three identical squares.
1Long side = 3 × short side.
2Draw the short side, a perpendicular, and step the compass (same opening) three times along it.
3Complete the rectangle and draw the two dividing lines.
A rectangle made of 3 equal squares: long side = 3 × short side (equal sides marked |).
✅ Answer: e.g. 3 cm × 9 cm
๐ Page 201Cannot be divided
Give side lengths of a rectangle that cannot be divided into (a) two identical squares (b) three identical squares.
1(a) Needs long side = 2 × short side. 4 cm × 6 cm does not work.
2(b) Needs long side = 3 × short side. 4 cm × 10 cm does not work.
✅ Answer: e.g. (a) 4 cm × 6 cm (b) 4 cm × 10 cm
๐ Page 201Construct · A square within a rectangle
In a rectangle 8 cm × 4 cm, construct a square with the same centre.
1The square fits the height: side = 4 cm.
2Space left = 8 − 4 = 4 cm, split equally: 2 cm on each side.
3Mark 2 cm from each short side on the top and bottom, and join.
4Check: the diagonals of both cross at the same point.
8 cm × 4 cm rectangle with a 4 cm square in the middle (2 cm from each short side).
✅ Answer: Square of side 4 cm, 2 cm from each short side.
๐ Textbook Page 202
๐ Page 202Construct · Falling Squares
Construct the falling squares (three 4 cm squares, then squares of 7, 5 and 3 cm).
1Draw the first square.
2Its top-right corner is the bottom-left corner of the next square; extend the top side to the right.
3Use perpendiculars and the compass to mark each side length.
Falling squares: each square starts where the previous one ends (4 cm, 4 cm, 4 cm).Falling squares: each square starts where the previous one ends (7 cm, 5 cm, 3 cm).
✅ Answer: See the figures.
๐ Page 202Construct · Shadings
Construct the shaded square figure with measurements of your choice.
1Big square of side 8 cm.
2Divide each side into 4 parts of 2 cm and draw the grid lines — but leave the bottom-left 4 cm × 4 cm block as one square.
3Draw one diagonal in each small square and shade one half with lines.
Shadings: a big square split into 4 × 4 small squares; one corner block of 2 × 2 is left as one square.
✅ Answer: e.g. big square 8 cm, small squares 2 cm.
๐ Textbook Page 203
๐ Page 203Construct · Square with a Hole
Draw a square with a circle whose centre is the centre of the square.
1Draw the square.
2Draw both diagonals lightly; they cross at the centre.
3Put the compass tip there and draw the circle (radius less than half the side).
Centre of the circle = point where the diagonals cross.
✅ Answer: Centre of circle = where the diagonals cross.
๐ Page 203Construct · Square with more Holes
Draw a square divided into 4 squares, each with a circle in the middle.
1Join the midpoints of opposite sides to make 4 small squares.
2Find the centre of each small square (its diagonals cross) and draw equal circles.
Split into 4 squares; each circle is centred where the small square’s diagonals cross.
✅ Answer: Four equal circles at the centres of the four small squares.
๐ Page 203Construct · Square with Curves
A square of side 8 cm has 4 equal curves, one on each side. Where is the compass tip placed?
1Each curve joins two corners and bulges into the square.
2Its centre is outside the square on the line through the middle of that side.
3Example: radius 5 cm, centre 3 cm outside the middle of the side (then it reaches both corners, since 3, 4, 5 make a right triangle).
4Repeat for all 4 sides.
Each arc: radius 5 cm, centre 3 cm outside the middle of a side; it passes through both corners.
✅ Answer: Centre outside, on the middle line of each side; e.g. radius 5 cm, 3 cm outside.
๐ Page 203Diagonals of a rectangle
Construct a rectangle PQRS, join PR and QS. Compare the lengths of the diagonals.
1Measure both diagonals.
2They are equal (for 7 cm × 4 cm, both ≈ 8.1 cm).
Rectangle 7 cm × 4 cm: d = e = a = h ≈ 30°, c = f = b = g ≈ 60°.
✅ Answer: The diagonals are equal.
๐ Textbook Page 204
๐ Page 204Angles made by the diagonals
Are g and h equal? Are c and d equal? Which angles are equal?
1In a rectangle that is not a square, g ≠ h and c ≠ d.
2Equal pairs: d = h = e = a and c = g = f = b (measure them).
3Each pair at a corner adds up to 90°.
✅ Answer: g ≠ h, c ≠ d (unless it is a square); d = e = a = h and c = f = b = g.
๐ Page 204Explore · Equal halves
How should the rectangle be made so that a diagonal divides the opposite angles into equal parts?
1Try a square: each angle is split into 45° and 45°.
Square: every angle is split into 45° + 45°.
✅ Answer: When all sides are equal — a square.
๐ Page 204Math Talk · Laws
What general laws did you observe? How can we be sure they always hold?
1Diagonals of a rectangle are equal.
2Each diagonal makes equal angles with opposite sides (alternate pairs).
3Only in a square does a diagonal cut the corner angles in half.
4Checking many rectangles makes us confident; later classes prove these with reasoning.
✅ Answer: Diagonals equal; square ⇔ diagonals bisect the angles.
๐ Textbook Page 205
๐ Page 205Example · Diagonal at 60° and 30°
Construct a rectangle in which one diagonal divides the opposite angles into 60° and 30°.
1Draw AB of any length and a perpendicular at B.
2At A draw a 60° angle; its arm meets the perpendicular at C.
3Draw a perpendicular at A (the line for D). The other part of ∠A is 90° − 60° = 30°.
4Find D: a perpendicular at C, or mark AD = BC with the compass. Join CD.
AB any length; angle 60° at A meets the perpendicular at B in C; then complete with D.
✅ Answer: See the construction; the other angle at A is 30°.
๐ Textbook Page 208
๐ Page 208Example · Side 5 cm, diagonal 7 cm
Construct a rectangle with one side 5 cm and a diagonal of 7 cm.
1Draw DC = 5 cm and the perpendicular l at C.
2B is on l and 7 cm from D.
3Draw an arc (part of a circle) of radius 7 cm with centre D; it cuts l at B.
4Draw perpendiculars at D and B; they meet at A.
Side 5 cm, diagonal 7 cm: arc of radius 7 cm from D cuts the perpendicular at C in B.
✅ Answer: See the construction (BC ≈ 4.9 cm).
๐ Textbook Page 211
๐ Page 211Construct · Q1
Construct a rectangle in which a diagonal divides the opposite angles into 50° and 40°.
1AB any length; perpendicular at B.
2Make 50° at A; it meets the perpendicular at C.
3Perpendicular at A; mark AD = BC; join CD.
4Check: ∠DAC = 90° − 50° = 40°.
AB any length; angle 50° at A meets the perpendicular at B in C; then complete with D.
✅ Answer: See the figure.
๐ Page 211Construct · Q2
Construct a rectangle where a diagonal divides the angles into 45° and 45°. What do you notice about the sides?
1Make 45° at A; it meets the perpendicular at B in C.
2Triangle ABC has two 45° angles, so AB = BC.
45° + 45°: the rectangle comes out as a square.
✅ Answer: All sides are equal — it is a square.
๐ Page 211Construct · Q3
Construct a rectangle with one side 4 cm and diagonal 8 cm.
1DC = 4 cm, perpendicular at C.
2Arc of radius 8 cm from D cuts it at B.
3Complete the rectangle; the other side measures about 6.9 cm.
Side 4 cm, diagonal 8 cm: arc of radius 8 cm from D cuts the perpendicular at C in B.
✅ Answer: See the figure.
๐ Page 211Construct · Q4
Construct a rectangle with one side 3 cm and diagonal 7 cm.
1DC = 3 cm, perpendicular at C.
2Arc of radius 7 cm from D cuts it at B.
3Complete the rectangle; the other side is about 6.3 cm.
Side 3 cm, diagonal 7 cm: arc of radius 7 cm from D cuts the perpendicular at C in B.
✅ Answer: See the figure.
๐ Page 2118.6 House
Recreate the house: all border lines are 5 cm, the door is 1 cm × 2 cm.
The house (all border lines 5 cm).
1Step 1: DE = 5 cm; perpendiculars at D and E; mark DB = EC = 5 cm. Draw the door 2 cm high and 1 cm wide in the middle.
2Step 2–3: A is 5 cm from both B and C: draw arcs of radius 5 cm from B and from C; they cross at A.
3Join AB and AC.
4Step 4: with radius 5 cm and centre A, draw the arc from B to C (the curved roof).
House with all border lines 5 cm: A is where arcs of radius 5 cm from B and C cross; the roof curve is an arc centred at A.
✅ Answer: See the house.
๐ Textbook Page 213
๐ Page 213Think · Full circles?
Was it necessary to draw two full circles to get A?
1Only the parts near A are used.
✅ Answer: No — two small arcs crossing at A are enough.
๐ Textbook Page 215
๐ Page 215Construct · Q1
Construct a bigger house in which all sides are 7 cm.
1Same steps with 7 cm: DE, DB, EC = 7 cm; arcs of radius 7 cm from B and C give A; roof arc of radius 7 cm from A.
House with all border lines 7 cm: A is where arcs of radius 7 cm from B and C cross; the roof curve is an arc centred at A.
✅ Answer: See the figure.
๐ Page 215Construct · Q2
Recreate ‘A Person’, ‘Wavy Wave’ and ‘Eyes’ using the ideas of the house construction.
1Person: the shoulder arc passes through the two top corners — find its centre with two equal arcs from the corners (like A in the house).
2Eyes: the centres A and B are equal distances above and below the corners’ line; find each by arcs of the same radius from both corners.
3Wavy wave: centres are the midpoints of AX and XB.
✅ Answer: Use ‘equal arcs from two points’ to find each centre.
๐ Page 215Construct · Q3
Is there a 4-sided figure with all sides equal that is not a square? Construct it.
1Draw AC = 5 cm and AB = 5 cm at any angle other than 90° (e.g. 60°).
2D must be 5 cm from both B and C: draw arcs of radius 5 cm from B and C — they cross at D.
3Join BD and CD. All sides are 5 cm but the angles are not 90°.
All four sides 5 cm, but the angles are not 90° — a rhombus, not a square.
✅ Answer: Yes — a rhombus.
๐ Where do we use this?
Rangoli and kolam designs, architecture drawings, tiles, logos, engineering drawings.
๐ Link to higher classes
Class 7: constructing triangles. Class 8: quadrilaterals and their properties.
CHAPTER 2 - Power Play nแต is n × n × n × n × ... × n (n multiplied by itself a times) and n⁻แต = 1 n a . nแต × nแต = nแต⁺แต (nแต)แต = (nแต)แต = n...
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