NCERT Solutions Ganita Prakash Chapter 8 Construct — Points Equidistant from Two Given Points
Book page 215 Updated on2026-09-05
Q1.
Construct a bigger house in which all the sides are of length 7 cm.
Answer
Same construction, with 5 cm replaced by 7 cm everywhere.
Draw DE = 7 cm. At D and E draw perpendiculars and cut off 7 cm to get B and C. Join BC lightly — the wall BCED is a square of side 7 cm.
Open the compass to 7 cm. With the tip at B draw an arc above the wall; with the tip at C draw another. They cross at A.
Join AB and AC. (Each is 7 cm.)
Keeping the opening at 7 cm, place the tip at A and draw the arc from B to C.
Draw the door — keep it in proportion, say 1.5 cm wide and 3 cm tall, standing on DE.
Every straight border: BD = DE = EC = AB = AC = 7 cm
radius used for all three arcs = 7 cm
Tip: only one number changes in the whole construction. That is the beauty of a construction plan — it works for any size.
Q2.
Try to recreate ‘A Person’, ‘Wavy Wave’, and ‘Eyes’ from the section ‘Artwork’, using ideas involved in the ‘House’ construction.
Answer
The one idea from the House — find a point by crossing two arcs, then draw the curve from it — solves all three pieces of artwork.
Artwork
Using the ‘House’ idea
A Person
Draw the body with corners B and C at the top. Fix a radius (say 3.5 cm) and cut two arcs of that radius from B and from C; they cross above the body at a point O. With the tip at O draw the arc from B to C — it dips into the body, just like the roof arc of the house, only drawn from the other side.
Wavy Wave
Mark A, X and B on the central line. For each wave, cut two arcs of the chosen radius from the two end points (A and X, then X and B); each crossing is the position for the compass tip. Put the tip above the line for one wave and below for the other, and draw with the same opening — the two waves come out identical.
Eyes
Mark the two corners of the eye. Cut arcs of the same radius from both corners — one crossing lies above the eye (point A) and the other below (point B). Draw the upper curve from B and the lower curve from A. Because the two crossings are mirror images, the eye is automatically symmetrical.
Why the same idea keeps working: in every one of these pictures a curve has to pass through two given points with a chosen radius. The centre of such a curve must be at that radius from both points — and the only way to find such a centre is to cross two arcs, exactly as we did for the apex A of the house.
Q3.
Is there a 4-sided figure in which all the sides are equal in length but is not a square? If such a figure exists, can you construct it?
Answer
Yes. It is a rhombus — a “pushed-over” square. All four sides are equal but the angles are not 90°, so property S2 fails and it is not a square. (Figures B and C on page 194 were rhombuses.)
Steps of construction (side 5 cm, as in Hint B of the book)
Draw AC = 5 cm.
From A draw another segment AB = 5 cm, slanting — not at a right angle to AC (say at about 70°).
Open the compass to 5 cm. With the tip at B draw an arc; with the same opening and the tip at C draw another. They cross at D, which is 5 cm from both B and C.
Join BD and CD. ABDC is the required figure.
A rhombus of side 5 cm. The point D is found exactly as the apex of the house was — by crossing two arcs of radius 5 cm from B and from C.
Why equal sides are not enough: S1 (equal sides) fixes only the lengths, not the corners. A four-sided figure with equal sides can still be tilted — push a square sideways and it becomes a rhombus, with two angles less than 90° and two more than 90°. To be a square it must satisfy S2 as well.
Try This: make a square from four equal drinking straws joined at the corners with thread, then push one corner. The straws stay the same length but the shape flattens — you have made a rhombus.