Question

An isosceles triangle PQR is inscribed inside a circle. If PQ = PR = $8\sqrt5$ cm and QR = 16 cm, then find the radius of the circle.

• A. 8 cm
• B. 12 cm
• C. 15 cm
• D. 10 cm

Answer 1

Answer: (D)

10 cm

Answer 2

We know circum-radius of a triangle = {product of sides/(4 × area of the triangle)}
For triangle PQR, let PM be the perpendicular bisector of QR

Therefore, PR = $8\sqrt5$ cm, MR = $\frac{16}{2}$ = 8 cm
In triangle PMR, using Pythagoras theorem
PM2 = PR2 - MR2
Or, PM2 = ($8\sqrt5$)2 - 82 = 320 - 64
Or, PM2 = 256
Or, PM = 16 (Since, length cannot be negative)
Therefore, area of the triangle = $(\frac{1}{2})$ × 16 × 16 = 128 cm2
Required circum-radius = $\frac{(8\sqrt5 × 8\sqrt5 × 16)}{(4 × 128)}$ = 10 cm
So, the correct option is (D) : 10 cm.