IMC 2018 Q4
4. A kite is made by joining two congruent isosceles triangles, as shown.
What is the value of \(x\)?
- A 36
- B 54
- C 60
- D 72
- E 80
Intermediate Mathematical Challenge 2018, Question 4. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 72
As \(PQ = PR\), \(\angle PRQ = \angle PQR = x°\). Also, as \(PR = PS\), \(\angle PRS = \angle PSR = x°\). Therefore, \(\angle QRS = (2x)°\). The sum of the interior angles of quadrilateral \(PQRS\) is 360°. Therefore \(x + x + 2x + x = 360\), so \(5x = 360\). Hence \(x = 72\).
Solution © UK Mathematics Trust, from the IMC 2018 solutions. Longer versions with investigations: Solutions and investigations (PDF).