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NCERT Solutions Ganita Prakash (Part 1) Chapter 4 Puzzle Time — Which Quad?

Book page 111 Updated on2026-09-05

Q1.
Fold a sheet into half.
Answer

Bring one edge exactly onto the opposite edge and press the crease flat.

The crease you make is the perpendicular bisector of the two side edges: it passes through their midpoints, and it meets them at 90° because the two halves lie exactly on top of each other.

Why folding gives an exact right angle: when the paper is folded, the part of an edge on one side of the crease lands precisely on the part on the other side. The two angles the edge makes with the crease are therefore equal, and together they make a straight angle — so each is 90°. No protractor could do better.
Q2.
Now, fold it once more into a quarter.
Answer

Fold the half-sheet in half again, this time across the first crease.

You now have a quarter sheet with two creases meeting at right angles, and their crossing point is the centre of the original sheet. Every corner of the quarter sheet is a right angle, and one of those corners is the centre point.

Original sheet → 2 halves → 4 quarters
Crease 1 ⊥ Crease 2, meeting at the centre of the sheet
Tip: keep track of which corner of the folded quarter is the centre of the sheet — that is the corner Step 3 asks you to fold, and it is what makes the final shape symmetric.
Q3.
Make a triangular crease at the corner that is at the middle of the paper.
Answer

Fold the centre corner over along a straight line joining a point on one edge to a point on the other. The corner turns down as a small triangle.

Because you are folding through four layers of paper at once, this single crease is copied into all four quarters when you open the sheet out.

Why one fold makes four creases: the quarter sheet is four layers thick, and the two earlier creases act as mirror lines. Whatever you do to the top layer is reflected across the vertical crease, across the horizontal crease, and across both — giving four copies of the same segment arranged symmetrically about the centre.
Q4.
Open the sheet. What is the shape formed by the creases?
Answer

The four copies of the triangular crease meet up to form a rhombus centred at the middle of the sheet.

4 congruent crease segments  ⇒ four equal sides  ⇒ a rhombus
Its diagonals lie along the two original creases, so they
  • bisect each other (they cross at the centre), and
  • meet at 90° (the first two folds were perpendicular)

If your triangular fold happened to cut off equal lengths on the two edges, the rhombus is a square; otherwise it is a leaning rhombus.

Why the four sides must be equal: the four crease segments are reflections of one another in the two perpendicular fold lines, and reflection preserves length. So the figure has four equal sides — which is exactly the definition of a rhombus.
Q5.
How would you fold the quarter paper to get the kinds of creases shown in the following image.
Answer

The picture shows a family of nested rhombuses, one inside the other, all sharing the same centre.

Make several triangular folds at the centre corner instead of one — folding the corner over a small amount, opening it, folding it over a larger amount, opening again, and so on.

  1. Fold the sheet into a quarter, as before.
  2. At the centre corner, fold a small triangle and press. Unfold.
  3. Fold a slightly larger triangle at the same corner, parallel to the first crease. Unfold.
  4. Repeat with steadily larger triangles.
  5. Open out the whole sheet.

Each fold gives one rhombus, and folds made parallel to one another give rhombuses of the same shape sitting neatly inside each other.

Why the rhombuses stay similar: the creases at the corner are parallel, so each cuts the two edges in the same ratio. The rhombuses they produce therefore have the same angles and differ only in size — the pattern shrinks towards the centre without changing shape.
Q6.
How would you fold the quarter paper such that a square is formed?
Answer

Make the triangular fold at the centre corner a 45° fold — that is, fold so that the crease cuts off equal lengths along the two edges meeting at the centre.

Cut off length d on each of the two perpendicular creases
The four crease segments each measure √(d² + d²) = d√2 ⇒ all sides equal
The diagonals of the figure are both 2d ⇒ equal diagonals
They bisect each other at 90°  (the original two folds)
⇒ equal diagonals, bisecting each other at 90° ⇒ a square

The easiest way to be exact: bring one of the two creases exactly onto the other at the centre corner. That fold bisects the right angle, so it makes 45° with each — and it automatically cuts off equal lengths.

Why unequal cuts fail: if you cut off d along one crease and a different length e along the other, the diagonals become 2d and 2e. They still bisect each other at 90°, so you still get a rhombus, but unequal diagonals mean unequal — not right — angles. Only d = e makes the diagonals equal, and only then is the rhombus a square.
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