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

The Correct Option is D

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.