NCERT Solutions Ganita Prakash Chapter 8 – 201Section 8.4 An Exploration in Rectangles — Construct: Breaking Rectangles
Book page 199 Updated on2026-09-05
Q1.
Breaking Rectangles. Construct a rectangle that can be divided into 3 identical squares as shown in the figure.
Answer
First plan with a rough diagram. If the small squares have side s, the rectangle is s tall and 3s long. So the long side must be three times the short side.
Take the short side as 3 cm, so the long side is 9 cm.
Draw AF = 3 cm (the short side, drawn vertically).
At A draw a perpendicular to AF, long enough to hold 9 cm.
Open the compass to AF (3 cm) and, starting from A, step it along the perpendicular three times to get B, then the next point, then C. Now AC = 9 cm.
At C draw a perpendicular to AC and cut off 3 cm to get D. Join FD.
Join the marks on AC to the matching marks on FD. The rectangle ACDF is now cut into three identical squares of side 3 cm.
A 9 cm × 3 cm rectangle falls into three identical 3 cm squares.
Why the long side must be 3 × the short side: the three squares sit side by side and each of them is as tall as the rectangle. So each square has side = the short side, and three of them laid in a row make a length of three times that.
Q2.
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
The way to plan is to draw a rough diagram of the finished figure first and mark on it everything that must be equal.
Rough diagram: rectangle ACDF, cut by the segment BE into the squares ABEF and BCDE.
Since the two squares are identical: AB = BC and FE = ED
Since ABEF is a square: AF = AB = BE = FE
Since BCDE is a square: BE = BC = CD = ED
So all the short segments are equal — and AC = 2 × AF
Steps of construction (taking the short side 3 cm):
Draw AF = 3 cm.
Draw a perpendicular to AF at A.
Open the compass to AF and step it twice along that perpendicular: first step gives B, second gives C. So AC = 6 cm.
At C draw a perpendicular, cut off 3 cm to get D, and join FD.
Join B to E (the matching point on FD). ABEF and BCDE are the two identical squares.
Tip: put a small tick mark ‘|’ on each segment that is equal to the others. That single habit turns a rough sketch into a construction plan.
Q3.
To draw the rectangle ACDF, one could assign any length to AF. For example, if we assign AF = 4 cm, then what must the length of AC be? Explore: Can the rectangle now be completed?
Answer
AF = 4 cm AB = BC = AF = 4 cm (all the short segments are equal) AC = AB + BC = 4 + 4 = 8 cm
Yes, the rectangle can now be completed, because both of its sides are known: 8 cm and 4 cm.
Draw AF = 4 cm; draw the perpendicular to AF at A.
Mark B at 4 cm and C at 8 cm along it.
Draw the perpendicular to AC at C and mark D with CD = 4 cm.
Join FD, and join BE. The rectangle 8 cm × 4 cm splits into two 4 cm squares.
Did you know? For three identical squares the same rule gives AC = 3 × 4 = 12 cm; for four squares, 16 cm. The rectangle 8 × 4 is exactly the one used in the next task, “A Square within a Rectangle”.
Q4.
In fact, one could proceed by drawing AF without even measuring its length using a ruler. As, AB = AF, we need to somehow transfer the length of AF to get the point B. How do we do it without a ruler? Can it be done using a compass?
Answer
Yes — this is exactly what a compass is best at. A compass can carry a length from one place to another without ever telling you what the length is.
Draw AF of any length and draw the perpendicular to AF at A.
Put the metal tip on A and open the compass until the pencil reaches F. Now the opening is the length AF.
Without changing the opening, keep the tip at A and cut the perpendicular. The cut is B, and AB = AF.
Shift the tip to B and cut again — that gives C, with BC = AB. (One more step gives a third square.)
Finish the rectangle with perpendiculars at C and the segment FD.
Why it works: the compass opening does not change when you move it, so every arc you draw has the same radius. Stepping it along a line therefore lays out equal lengths one after another — like using a pair of dividers instead of a ruler.
Check it yourself: measure AF, AB and BC at the end with a ruler. All three come out the same, even though you never measured while constructing.
Q5.
Give the lengths of the sides of a rectangle that cannot be divided into — two identical squares; three identical squares.
Answer
A rectangle splits into n identical squares exactly when its long side is n times its short side. So choose lengths that break this rule.
Rectangle
Is long = 2 × short?
Is long = 3 × short?
Conclusion
4 cm × 2.5 cm
No (2 × 2.5 = 5)
No (3 × 2.5 = 7.5)
cannot be divided into 2 or 3 identical squares
7 cm × 2 cm
No (2 × 2 = 4)
No (3 × 2 = 6)
cannot be divided into 2 or 3 identical squares
5 cm × 3 cm
No (2 × 3 = 6)
No (3 × 3 = 9)
cannot be divided into 2 or 3 identical squares
6 cm × 3 cm
Yes
No
splits into 2 squares, but not into 3
9 cm × 3 cm
No
Yes
splits into 3 squares, but not into 2
Answers: a rectangle of 4 cm × 2.5 cm (or 5 cm × 3 cm) cannot be divided into two identical squares; a rectangle of 7 cm × 2 cm (or 5 cm × 3 cm) cannot be divided into three identical squares. Many more are possible — try your own.
Why: if n identical squares fill the rectangle in a row, each square must be as tall as the rectangle, so its side equals the short side; the total length is then n times the short side. If the long side is not that exact multiple, no such division exists.