PK 2019 Q10
10. A square has vertices \(P\), \(Q\), \(R\), \(S\) labelled clockwise. An equilateral triangle is constructed with vertices \(P\), \(T\), \(R\) labelled clockwise. What is the size of angle \(RQT\) in degrees?
- A 30
- B 45
- C 135
- D 145
- E 150
Pink Kangaroo 2019, Question 10. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 135
The diagram of the square and the triangle is shown. Consider the triangles \(PQT\) and \(RQT\). They share side \(QT\); also \(PQ = RQ\) because they are sides of a square; and \(PT = RT\) because they are sides of the equilateral triangle, so by SSS the triangles \(PQT\) and \(RQT\) are congruent. Therefore \(\angle PQT = \angle RQT\). Also \(\angle PQT + \angle RQT + 90° = 360°\) so \(\angle RQT = \tfrac{1}{2}(360° - 90°) = 135°\).
Solution © UK Mathematics Trust, from the PK 2019 solutions.