NCERT Solutions Ganita Prakash Chapter 8 Construct — Exploring Diagonals of Rectangles and Squares
Book page 211 Updated on2026-09-05
Q1.
Construct a rectangle in which one of the diagonals divides the opposite angles into 50° and 40°.
Answer
Note first that 50 + 40 = 90, so such a rectangle is possible.
Steps of construction
Draw AB = 5 cm (any convenient length).
At B draw a perpendicular to AB.
At A draw a ray making 50° with AB. Mark C where it meets the perpendicular.
At A draw a perpendicular to AB. At C draw a perpendicular to BC. They meet at D.
Join CD. ABCD is the required rectangle; the diagonal AC divides ∠A into 50° and 40°, and ∠C in the same way.
Check: 50° + 40° = 90° = ∠A ✔ with AB = 5 cm, BC comes out ≈ 6 cm
Tip: mark the 50° carefully. If the ray is even 2° out, the second part will not measure 40° and the rectangle will refuse to close neatly.
Q2.
Construct a rectangle in which one of the diagonals divides the opposite angles into 45° and 45°. What do you observe about the sides?
Answer
Steps of construction
Draw AB = 5 cm. At B draw a perpendicular to AB.
At A draw a ray making 45° with AB; it cuts the perpendicular at C.
Complete the rectangle as before to get D.
Observation: all four sides come out equal — the rectangle is a square.
AB = 5 cm and BC = 5 cm so AB = BC = CD = DA → it is a square the diagonal AC ≈ 7.1 cm
Why it happens: a diagonal that makes 45° with AB is leaning equally towards both sides, so it must rise as much as it runs: BC = AB. The moment two adjacent sides are equal, all four are equal and the rectangle becomes a square. This is the answer to the “Explore” question of page 204.
Q3.
Construct a rectangle one of whose sides is 4 cm and the diagonal is of length 8 cm.
Answer
Steps of construction
Draw the base DC = 4 cm.
At C draw a perpendicular to DC — call it l.
Open the compass to 8 cm, put the tip at D and draw an arc cutting l at B.
Draw a perpendicular to DC at D and a perpendicular to CB at B; they meet at A. Join the figure.
The other side CB works out as: 4 cm across, 8 cm slant
CB ≈ 6.9 cm
Check: DC = AB = 4 cm, CB = DA ≈ 6.9 cm, all angles 90°, and the second diagonal AC also measures 8 cm.
Tip: the diagonal must always be longer than either side. With a side of 4 cm, any diagonal shorter than 4 cm would be impossible — the arc would never reach the line l.
Q4.
Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.
Answer
Steps of construction
Draw DC = 3 cm.
At C draw the perpendicular l to DC.
With the tip at D and radius 7 cm, draw an arc cutting l at B.
Complete the rectangle with perpendiculars at D and B, meeting at A.
other side CB ≈ 6.3 cm
check: DC = AB = 3 cm, CB = DA ≈ 6.3 cm, all angles = 90°
Why the second side is not given: a side and a diagonal are enough to pin the rectangle down. Once DC and the diagonal DB are fixed, the right angle at C leaves only one possible position for B — so CB cannot be chosen, it is decided for you.