Euclidean geometry — Grade 12 Maths Past Paper Questions

Triangle KLM with P on KL and Q on KM such that PQ is parallel to LM, and PM and LQ drawn. Prove that KP/PL = KQ/QM — the proportion theorem, which is one of the proofs that can be asked for by name in the examination.

SBT, SA and TC are tangents to a circle at B, A and C, with AB produced to P and AC produced to Q so that T lies on PQ, and AB/AP = AC/AQ. Prove an angle equality, prove that triangles ABC and TCQ are similar, prove that ABTQ is a cyclic quadrilateral, and prove that TQ is a tangent to circle TBC at T.

M is the centre of the circle through A, B and C, E lies on AC, AC bisects one angle and EB bisects another, and a third angle is x. Determine a second angle in terms of x, show that angle BAC = 90° − 2x, and prove that AE is a diameter of circle ABE.

XMN is a straight line, XT is a tangent to the circle, and Y lies on XN with XY = XT. Prove that YT bisects angle MTN and that XM/XT = XT/XN. Then, with MY = 20 mm, YN = 50 mm and XT = k mm, express XM in terms of k and calculate the length of k.