Maths Olympiad - Triangles 1

Question:

PQ = 40cm, RS = 20cm, PS = 60cm and Angle QPS = Angle RSP = 60 degrees. Find Angle QRS.

Answer:

Draw line RX parallel to PS. Since Angle QPS = Angle RSP = 60 degrees, then Angle QXR = 60 degrees and line QX = XP = 20cm.

Angle PYS = 180 - (QPS + RSP) = 60 degrees

Therefore, Triangle YPS is equilateral because all of its angles are 60 degrees.

Since, the lengths of all sides are equal in an equilateral triangle, PY = SY = PS = 60cm.

Therefore, QY = 20cm (Diagram not drawn to scale) and Q is the midpoint of line XY.

Therefore, line QR is a bisector of Angle XRY = 60 degrees which makes Angle QRX = 60 / 2 = 30 degrees.

Angle XRS = 180 - PSR (internal angles) = 120 degrees.

Therefore, Angle QRS = QRX + XRS = 150 degrees.