NCERT Solutions Ganita Prakash (Part 1) Chapter 7 –151Section 7.2 Constructing a Triangle When its Sides are Given — Figure it Out
Book page 150 Updated on2026-09-05
Q1.
Use the points on the circle and/or the centre to form isosceles triangles.
Answer
Take the centre O and any two points P and Q on the circle. ∆OPQ is isosceles.
OP and OQ are both radii, so ∆OPQ is always isosceles.
OP = radius OQ = radius So OP = OQ → ∆OPQ is isosceles
Why it happens: Every point of a circle is the same distance from the centre. So the moment two of the three vertices are on the circle and the third is the centre, two sides are forced to be equal. You can slide P and Q anywhere on the circle and the triangle stays isosceles.
Try This: Mark P, Q, R, S … on the circle. ∆OPQ, ∆OQR, ∆ORS … are all isosceles — you can make as many as you like. If you place P and Q so that PQ also equals the radius, the triangle even becomes equilateral.
Q2.
Use the points on the circles and/or their centres to form isosceles and equilateral triangles. The circles are of the same size.
Answer
In both figures each circle passes through the centre of the other, so AB = radius.
P and Q are the crossing points of the two equal circles; ∆ABP and ∆ABQ are equilateral.
Equilateral triangles
AP = radius (P is on circle A) BP = radius (P is on circle B) AB = radius (each circle passes through the other's centre) So AP = BP = AB → ∆ABP is equilateral (and so is ∆ABQ)
Isosceles triangles
Take any point C on the circle with centre B. Then BC = radius = AB, so ∆ABC is isosceles. Sliding C round that circle gives endlessly many.
Take any two points C and D on the circle centred at A. Then AC = AD = radius, so ∆ACD is isosceles.
∆APQ and ∆BPQ are isosceles too, since AP = AQ and BP = BQ.
In the second figure A, B and C are the three centres, and each of them lies on the other two circles:
AB = BC = CA = radius So ∆ABC is equilateral
Why it happens: Equal circles fix one single length — the common radius. Every segment you can draw from a centre to a point of its own circle equals that length. Counting how many of a triangle's sides are such segments tells you at once whether it is isosceles (two of them) or equilateral (all three).