Question

3.1 Complete the following statements so that they are true: 3.1.1The angle between the tangent and chord is ... 3.1.2Opposite angles of a cyclic quadrilateral are ... 3.2In the diagram below, two circles have a common tangent TAB. PT is a tangent to the smaller circle. PAQ, QRT and NAR are straight lines. Let Q = 40°. N B 40° 3.2.1 Determine, with reasons, THREE other angles equal to 40°. 3.2.2 If P₁= A4 prove that PTRN is a parallelogram. ​

Answer 1

3.1.1 The angle between the tangent and chord is equal to the angle subtended by the chord in the alternate segment.

3.1.2 Opposite angles of a cyclic quadrilateral are supplementary.

3.2.1 Three other angles equal to 40° are:

- Angle APT: Since tangent PT is perpendicular to radius OA, angle APT is equal to angle OAP which is the same as angle QAT (alternate angles).
- Angle QAR: Since triangle QAR is isosceles (QA = QR), angle QAR is equal to angle QRA which is equal to 40°.
- Angle QPN: Angle QPN is the angle between the tangent PT and chord QN. Using the result from 3.1.1, angle QPN is equal to the angle subtended by chord QR in the alternate segment which is equal to angle QAR.

3.2.2 We know that angle QAR = 40° and angle QRT = 90° (since RT is a radius and hence perpendicular to tangent PT). Therefore, angle PRT = 50° (sum of angles in triangle PRT is 180°).

Since angles PRT and NRT are alternate angles, they are equal. Therefore, angle NRT = 50°.

Since angles QNR and QTR are alternate angles, they are equal. Therefore, angle QNR = 90° - 50° = 40°.

Since angle QNR = angle QAR, quadrilateral QNAR is cyclic. Therefore, angle QNA = 180° - angle QAR = 140° (opposite angles of a cyclic quadrilateral are supplementary).

Finally, since angles PTR and NRT are equal and opposite angles of a parallelogram, quadrilateral PTRN is a parallelogram.

Hence, we have proved that PTRN is a parallelogram.