IMC 2017 Q15
15. In the diagram shown, \(PQ = SQ = QR\) and \(\angle SPQ = 2 \times \angle RSQ\).
What is the size of angle \(QRS\)?
- A 20°
- B 25°
- C 30°
- D 35°
- E 40°
Intermediate Mathematical Challenge 2017, Question 15. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 30°
Let \(\angle QRS = x\)°. Then, as \(SQ = QR\), \(\angle RSQ = x\)° also. Now \(\angle SPQ = 2 \times \angle RSQ = 2x\)°. In triangle \(PQS\), \(PQ = SQ\), so \(\angle PSQ = \angle SPQ = 2x\)°. Therefore \(\angle PSR = \angle PSQ + \angle RSQ = 2x\)° + \(x\)° = \(3x\)°. The sum of the interior angles of a triangle is 180°. So, considering triangle \(PSR\), \(\angle SPR + \angle PSR + \angle PRS = 180\)°. Therefore \(2x\)° + \(3x\)° + \(x\)° = 180°.
So \(6x = 180\), that is \(x = 30\).
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).