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NCERT Solutions Ganita Prakash Chapter 8 Figure it Out — Squares and Rectangles

Book page 194 Updated on2026-09-05

Q1.
Draw the rectangle and four squares configuration (shown in Fig. 8.3) 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

Steps on dot paper

  1. First draw the rectangle in the middle — say 6 dots long and 4 dots high, with its corners on dots.
  2. Now go out diagonally from each corner of the rectangle by the same number of dots (leave one dot gap in both directions).
  3. From that dot, draw a small square of side 3 dots so that its inner corner points at the corner of the rectangle.
  4. Repeat at all four corners, always leaving the same gap and using the same size of square.

What makes it symmetric: equal treatment of all four corners — same gap, same square size, same diagonal direction. Counting dots does this automatically, without any measuring.

Check it yourself: fold the drawing along the vertical middle line and then along the horizontal middle line. In a correct figure the four squares fall exactly on one another.
Q2.
Identify if there are any squares in this collection. Use measurements if needed.
Answer

Only A is a square. Read the corners off the dot grid — that is quicker and safer than measuring.

FigureHalf-diagonals (in dot steps)SidesAnglesWhat it is
A4 across, 4 downall four equalall 90°a square (a rotated one)
B3 across, 4 downall four equalnot 90°a rhombus, not a square
C5 across, 4 downall four equalnot 90°a rhombus, not a square
Dsides 1 right 2 down, and 2 right 1 uptwo long, two shortall 90°a rectangle, not a square
A ✔ B ✘ C ✘ D ✘ square rhombus rhombus rect.
Only A has equal sides and right angles, so only A is a square. B and C have equal sides but slanted corners; D has right angles but unequal sides.
Why B and C fail: in A the two diagonals are equally long (4 + 4 across and 4 + 4 down), so the four sides meet at 90°. In B the diagonals are 6 and 8 dot steps, in C they are 10 and 8 — the corners get squeezed and the angles are no longer right angles, even though all four sides are still of equal length.
Q3.
Think: Is it possible to reason out if the sides are equal or not, and if the angles are right or not without using any measuring instruments in the above figure? Can we do this by only looking at the position of corners in the dot grid?
Answer

Yes — the dots alone are enough. No ruler, no protractor.

  • For equal sides: describe each side as a move on the grid, e.g. “4 right and 4 down”. Two sides are equal if they need the same pair of steps (in some order). In A every side is a “4 and 4” move, so all four sides are equal.
  • For right angles: a side that goes “4 right, 4 down” meets a side that goes “4 right, 4 up” — turning one move sideways gives exactly the other. Whenever one move is the other one turned by a quarter turn, the corner is a right angle.
  • In D the moves are “2 right, 1 up” and “1 right, 2 down” — one is the other turned by a quarter turn, so D does have right angles, but the moves are of different sizes, so its sides are unequal.
Why this works: the dots are equally spaced in both directions, so counting steps is measuring — with the dot spacing as the unit. Equal step-pairs mean equal lengths, and a quarter-turn of a step-pair means a quarter-turn of the line, that is 90°.
Q3.
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

Use the “step” trick: pick a move, then turn it by a quarter turn for the next side.

FigureSteps for the sides (repeat the pattern round)Check
Rotated square 13 right 1 down, then 1 left 3 down, then 3 left 1 up, then 1 right 3 upall sides equal, all angles 90° ✔
Rotated square 22 right 2 down, 2 left 2 down, 2 left 2 up, 2 right 2 upall sides equal, all angles 90° ✔
Rotated square 34 right 1 down, 1 left 4 down, 4 left 1 up, 1 right 4 upall sides equal, all angles 90° ✔
Rotated rectangle 12 right 1 down, then 1 left 2 down — repeat twice as long on the long sides (4 right 2 down)opposite sides equal, all angles 90° ✔
Rotated rectangle 23 right 1 down for the long side, 1 left 3 down for the short sideopposite sides equal, all angles 90° ✔

How to verify: measure the four sides with a ruler (equal for a square; equal in pairs for a rectangle) and check each corner with the corner of a set square or a piece of paper — every corner must fit it exactly.

Tip: to turn a move by a quarter turn, swap the two numbers and change one direction: “3 right 1 down” becomes “1 left 3 down”. Do this four times and you come back to the start — the figure closes by itself.
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