NCERT Solutions Ganita Prakash Chapter 8 – 199Section 8.4 An Exploration in Rectangles — In-text Questions
Book page 197 Updated on2026-09-05
Q1.
Construct a rectangle ABCD with AB = 7 cm and BC = 4 cm. Imagine X to be a point that can be moved anywhere along the side AD. Similarly, imagine Y to be a point that can be moved anywhere along the side BC.
Answer
Steps of construction
Draw AB = 7 cm as the top side.
At A and at B draw perpendiculars to AB, going downwards.
Open the compass to 4 cm; from A cut the first perpendicular at D, from B cut the second at C.
Join DC — ABCD is the rectangle. (AD = BC = 4 cm, DC = AB = 7 cm.)
Mark any point X on AD and any point Y on BC, and join XY with a dashed line.
Rectangle ABCD (7 cm × 4 cm). X slides along AD, Y slides along BC, and we watch the length XY.
Tip: draw the rectangle four or five times on one page, so that you can try different positions of X and Y without rubbing out.
Q2.
At which positions will the points X and Y be at their closest? When do you think they will be the farthest? What does your intuition say? Discuss with your classmates.
Answer
Closest: when X and Y are at the same distance from A and from B — that is, when XY is parallel to AB. Then
XY = AB = 7 cm (the smallest possible value)
Farthest: when X and Y are at opposite ends of their sides — X at A with Y at C, or X at D with Y at B. Then XY is a diagonal of the rectangle.
XY = AC = BD = about 8.1 cm (the largest possible value)
Position of X
Position of Y
Length XY
X at A
Y at B
7 cm (smallest)
1 cm from A
1 cm from B
7 cm (smallest)
X at D
Y at C
7 cm (smallest)
X at A
Y at C
8.1 cm (largest)
X at D
Y at B
8.1 cm (largest)
Why it happens: going from X to Y you must cross the full width of the rectangle, 7 cm. If X and Y are level with each other you cross straight, and 7 cm is all you travel. If one of them is higher than the other you must also climb, and a slanting path is always longer than the straight crossing. The biggest climb possible is the full 4 cm — that gives the diagonal.
Q3.
How does the minimum distance between the points X and Y compare to the length of AB?
Answer
They are exactly equal.
minimum XY = AB = 7 cm
The smallest XY can ever be is the width of the rectangle, and that width is AB itself.
Why: AD and BC are opposite sides of a rectangle, so they are parallel and always 7 cm apart. XY joins a point of one to a point of the other, so XY can never be shorter than 7 cm — and it becomes exactly 7 cm when XY is drawn straight across, parallel to AB.
Q4.
How will you keep track of the lengths XY for different positions of X and Y? Is there a shorthand way of writing it down? (When X is 5 mm away from A and Y is 3 cm away from B, XY = ___ cm ___ mm; when X is 1 cm away from A and Y is 1 cm away from B, XY = ___ cm ___ mm; when X is 2 cm away from A and Y is 4 cm away from B, XY = ___ cm ___ mm.)
Answer
Yes — instead of writing a whole sentence each time, make a table with one column for each thing that changes.
Distance of X from A
Distance of Y from B
Length of XY
5 mm
3 cm
7 cm 4 mm
1 cm
1 cm
7 cm 0 mm
2 cm
4 cm
7 cm 3 mm
Measure each XY in your own figure — you should get these readings (correct to the nearest millimetre).
Why the middle row is exactly 7 cm: X and Y are both 1 cm below the top, so XY runs straight across, parallel to AB, and equals AB = 7 cm. In the other two rows there is a difference in height (2.5 cm in the first row, 2 cm in the third), so the line slants and comes out a few millimetres longer.
Tip: a table like this is a mathematician's shorthand. Only three things change, so three columns are enough — the rest of the sentence is written once, in the headings.
Q5.
Have you checked what happens to the length XY when X and Y are placed at the same distance away from A and B, respectively? (5 mm and 5 mm; 1 cm and 1 cm; 1 cm 5 mm and 1 cm 5 mm.) In each of these cases, observe 1. how the length XY compares to that of AB and 2. the shape of the 4-sided figure ABYX.
Answer
Distance of X from A
Distance of Y from B
Length of XY
Shape of ABYX
5 mm
5 mm
7 cm
a rectangle
1 cm
1 cm
7 cm
a rectangle
1 cm 5 mm
1 cm 5 mm
7 cm
a rectangle
1. In every such case XY = AB = 7 cm.
2. The 4-sided figure ABYX is always a rectangle — a small one at the top of the big rectangle.
Why it happens: AX and BY are equal parts of two equal, parallel sides, so X and Y stay level with each other. ∠A and ∠B are right angles because they are corners of the big rectangle, and XY crosses the two upright sides at right angles too. Four right angles and equal opposite sides — that is a rectangle, and its side XY must equal its opposite side AB.
Check it yourself: as X slides down from A to D with Y always level with it, the small rectangle ABYX grows taller and taller, but its width never changes. When X reaches D and Y reaches C, ABYX becomes the whole rectangle ABCD.
Q6.
How does the farthest distance between X and Y compare with the length of AC? BD?
Answer
The farthest distance is exactly equal to AC, and also to BD — the two diagonals of the rectangle.
farthest XY = AC = BD ≈ 8.1 cm
(check: 7 cm across and 4 cm down gives about 8.1 cm)
Why it happens: X can go no further than the ends A and D of its side, and Y no further than the ends B and C of its side. The longest join is from one end to the opposite end — X at A with Y at C gives the line AC, and X at D with Y at B gives DB. Those are the diagonals, and in a rectangle the two diagonals are always equal in length.
Check it yourself: draw both diagonals of your rectangle and measure them. Both come out about 8.1 cm, and they cut each other exactly in half.