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NCERT Solutions Ganita Prakash Chapter 8 & 204Section 8.5 Exploring Diagonals of Rectangles and Squares — In-text Questions

Book page 203 Updated on2026-09-05

Q1.
Consider a rectangle PQRS. Join PR and QS. Compare the lengths of the diagonals. First predict the answer. Then construct a rectangle marking the points as shown and measure the diagonals.
Answer

Prediction and result agree: the two diagonals of a rectangle are equal in length.

Construct a rectangle, say PQRS with PQ = 7 cm and QR = 4 cm, join PR and QS and measure:

PR = QS ≈ 8.1 cm
RectangleDiagonal 1Diagonal 2Equal?
7 cm × 4 cm8.1 cm8.1 cmYes
6 cm × 4 cm7.2 cm7.2 cmYes
5 cm × 5 cm (a square)7.1 cm7.1 cmYes
Why it happens: each diagonal joins two opposite corners, and both diagonals span the same width and the same height of the rectangle — one goes “7 across and 4 down”, the other “7 across and 4 up”. Same journey, only mirrored, so the two lengths must be the same.
Check it yourself: the diagonals also cut each other exactly in half. Measure the four halves — all four are equal.
Q2.
Observe that a diagonal divides each of the pair of opposite angles into two smaller angles. The diagonal PR divides angle R into g and h, and angle P into c and d. Are g and h equal? Are c and d equal? First predict the answers, and then measure the angles. What do you observe? Identify pairs of angles that are equal.
Answer

Take the rectangle 7 cm long and 4 cm high and measure the eight angles with a protractor:

c ≈ 60°, d ≈ 30°, e ≈ 30°, f ≈ 60°, g ≈ 60°, h ≈ 30°, a ≈ 30°, b ≈ 60°
each corner: 60° + 30° = 90° ✔

So g and h are NOT equal, and c and d are NOT equal — the diagonal does not cut the right angle in half (unless the rectangle happens to be a square).

The equal pairs are:

a = d = e = h (the four “flat” angles, ≈ 30°)
b = c = f = g (the four “steep” angles, ≈ 60°)
P Q R S d c e f b a g h
The two diagonals of rectangle PQRS make eight angles. The four flat ones (a, d, e, h) are equal, and so are the four steep ones (b, c, f, g).
Why: the diagonal PR is a slanting line that rises the same way at both of its ends, so the angle it makes with the long side at P equals the angle it makes with the long side at R — that is d = h. Because the long side is 7 cm and the short side only 4 cm, the diagonal leans much closer to the long side, so the “flat” angle is small and the “steep” angle is large. They can only be equal when the two sides are equal.
Q3.
Explore: How should the rectangle be constructed so that the diagonal divides the opposite angles into equal parts?
Answer

The rectangle must be constructed with its two adjacent sides equal — that is, it must be a square.

In a square: each corner = 90°
diagonal divides it into 45° + 45° = two equal parts
Why only a square works: the diagonal leans towards the longer side. If the length is bigger than the breadth, the diagonal makes a small angle with the length and a big angle with the breadth. The two parts become equal only when there is no “longer side” at all, i.e. when length = breadth.
Check it yourself: construct a square of side 6 cm, draw a diagonal and measure the two parts of a corner. Both read 45°.
Q4.
In your experimentation, did you consider the case when all four sides of the rectangle are equal? That is, did you consider the case of a square? See what happens in this special case!
Answer

In a square something special happens to all eight angles at once:

  • Every one of the eight small angles becomes 45°.
  • The two diagonals are equal and they cross each other at 90° — they are perpendicular.
  • Each diagonal cuts the square into two identical halves that fold exactly on to each other.
a = b = c = d = e = f = g = h = 45°
angle between the diagonals = 90°
Why: in a square the length and the breadth are the same, so the diagonal leans equally towards both sides and splits each 90° corner into two 45° parts. In a rectangle that is not a square, the diagonals still bisect each other but they meet at a slanted angle, not at 90°.
Q5.
What general laws did you observe with respect to the angles and sides? Try to frame and discuss them with your classmates. How can one be sure if the laws that you have observed will always be true?
Answer

Laws you can frame from the experiments

  1. The two diagonals of a rectangle are equal in length.
  2. The diagonals cut each other exactly in half.
  3. A diagonal splits each of the two opposite right angles into two parts, and the four “flat” parts are equal to each other, as are the four “steep” parts.
  4. The two parts of a corner are equal (45° each) only when the rectangle is a square.
  5. In a square the diagonals also meet at right angles.

How can we be sure? Measuring can never prove a law — a ruler and a protractor can only tell us that it holds for the few figures we drew, and always with a small error. To be sure we must give a reason that does not depend on the figure. For example:

A reason for law 1: fold or turn the rectangle by half a turn about the point where the diagonals cross. The rectangle falls exactly on itself, and the diagonal PR falls on QS. Two lines that fall on each other must be of the same length. Since this argument uses only the properties R1 and R2, it works for every rectangle, big or small — that is what makes it a law.
Math Talk: Discuss in class — “measuring ten rectangles” and “giving one reason” are very different things. In mathematics only the second one settles the matter.
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