The document Long Answer Type Questions- Circles Class 9 Notes | EduRev is a part of the Class 9 Course Class 9 Mathematics by VP Classes.

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**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, OP^{2} = PT^{2} + OT^{2}

⇒ r^{2 }= 6^{2} + (r - 3)^{2}

⇒ r^{2} = 36 + r^{2 }+ 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,

AD^{2} = AB^{2} - BD^{2}

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

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