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NCERT Solutions Ganita Prakash (Part 1) Chapter 4 .5 Playing with Quadrilaterals — Geoboard Activity

Book page 1034 Updated on2026-09-05

Q1.
Place two rubber bands perpendicular to each other, forming diagonals of equal length. Join the ends. What is the quadrilateral that you get? Justify your answer.
Answer

You get a square.

On the geoboard the two bands are stretched between pegs so that they cross at the middle peg — that is, each band is cut in half by the other.

Diagonals equal ✓  +  bisect each other ✓  ⇒ all four angles are 90°  (Deduction 3)
Diagonals meet at 90° ✓  ⇒ all four sides are equal  (Deduction 5)
⇒ a square
Why all four sides come out equal: the four small triangles round the centre each have two legs of the same length (half a band) with a right angle between them. So they are congruent by SAS, and their hypotenuses — the four sides of the figure — must be equal.
Check it yourself: if each band spans 6 units, then each half is 3 units and each side of the square measures √(3² + 3²) = 3√2 ≈ 4.24 units.
Q2.
Extend one of the diagonals on both sides by 2 cm. What quadrilateral will you get now? Justify your answer.
Answer

You get a rhombus — no longer a square, but still with four equal sides.

Extending one band by 2 cm at each end keeps its midpoint exactly where it was. So two things are unchanged and one thing changes:

ConditionBeforeAfter extending
Diagonals bisect each otherYesYes — the midpoint did not move
Diagonals perpendicularYesYes — the directions did not change
Diagonals equalYesNo — one is now 4 cm longer

Diagonals that bisect each other at right angles give a rhombus; the “equal” condition, which is what makes a rhombus a square, has been lost.

Why the sides stay equal: call the half-lengths p and q. Each of the four triangles at the centre is right-angled with legs p and q, so every side of the quadrilateral measures √(p² + q²). Making p larger changes that common value but keeps all four sides identical.
Tip: this is the geoboard version of the fact that a square is a rhombus with equal diagonals. Stretch one diagonal and the square relaxes into a leaning rhombus.
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