Question

A2 = 2.2 TR is a tangent to the circle at R in the diagram alongside. TQP is a straight line. RQ = RT, TÔR = x and PÔR = y. 2.2.1 Give, with reasons, THREE other angles equal to x. 2.2.2 If x= 75°, prove that PTRO is NOT a cyclic quadrilateral. ​

Answer 1

2.2.1 Answer: Three other angles equal to x are:

Angle QRP: Since RQ = RT, triangle RQT is an isosceles triangle and angle QRP is equal to angle TRQ (x) by the Isosceles Triangle Theorem.

Angle QPR: Since TR is a tangent to the circle at R and OR is a radius, angle TRQ (x) is equal to angle QPR by the Alternate Segment Theorem.

Angle QPO: Since TQP is a straight line, angle QPO is supplementary to angle QPR and therefore also equal to x.

2.2.2 Answer: If x = 75°, then angle QRP = 75° and angle QPR = 75°. Since TQP is a straight line, angle QPO = 180° - 75° = 105°. In order for PTRO to be a cyclic quadrilateral, opposite angles must add up to 180°. However, angle TRO + angle TPO = 75° + 105° = 180° ≠ 180°. Therefore, PTRO is not a cyclic quadrilateral.