IMC 2007 Q20
20. \(P\), \(Q\), \(R\) are points on the circumference of a circle of radius 4 cm. \(\angle PQR\) = 45°. What is the length of chord \(PR\)?
- A 4 cm
- B \(3\sqrt{3}\) cm
- C \(4\sqrt{2}\) cm
- D 5 cm
- E 6 cm
Intermediate Mathematical Challenge 2007, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C \(4\sqrt{2}\) cm
Let \(O\) be the centre of the circle. Then \(\angle POR\) = 90° as the angle subtended by an arc at the centre of a circle is twice the angle subtended by that arc at a point on the circumference of the circle. So triangle \(POR\) is an isosceles right-angled triangle with \(PO = RO = 4\) cm. Let the length of \(PR\) be \(x\) cm. Then, by Pythagoras' Theorem, \(x^2 = 4^2 + 4^2 = 2 \times 4^2\) and so \(x = 4\sqrt{2}\).
Solution © UK Mathematics Trust, from the IMC 2007 solutions.