Long Answer Type Questions: Circles - Mathematics (Maths) Class 9

Question 1. Look at the adjoining figure. If O is the centre of the circle and ST = 3 cm, then find the radius of the circle when RS ⊥ PQ.
 Solution: Let us join O and P such that OP = r.
∵ RS ⊥ PQ
∴ T is the mid-point of PQ.

⇒ PT = (1/2) PQ
⇒ PT = (1/2) x 12 cm = 6 cm                 [∵ PQ = 12 cm      (Given)]
And ∠ OTP = 90°
Also OS = r and TS = 3 cm
∴ OT = OS - TS = (r - 3) cm
Now, in right ΔOTP, we have, OP² = PT² + OT²
⇒ r² = 6² + (r - 3)²
⇒ r² = 36 + r² + 9 -  6r
⇒ 6r = 45

⇒ r =(45/6) = (15/2) = 7.5 cm
Thus, the radius of the circle is 7.5 cm.

Question 2. An equilateral triangle is inscribed in a circle. Find the radius of the circle.
 Solution: Let ABC be an equilateral triangle such that AB = BC = AC = 9 cm                  (each)
Let us draw a median AD corresponding to BC.
∴ BD =(1/2) BC
⇒ BD = (1/2) x 9 cm = (9/2)cm
Also, AD ⊥ BC                  [∵ O is the centre of the circle]
Now, in right ΔADB,

AD² = AB² - BD²

Since, in an equilateral triangle, the centroid and circumcentre coincide.

∴ AO: OD = 2:1

⇒  

⇒ Radius = 3√3 cm

Thus, the required radius =  3√3 cm