4 (a) North 114° A NOT TO SCALE C B A, B and C are three towns and the bearing of C from A is 114°. B is due south of A and AC = BC. Calculate the bearing of B from C. ................................................. [3] (b) R S 74° NOT TO SCALE Q 58° 27° M P N P, Q, R and S lie on a circle. MPN is a tangent to the circle at P. Angle MPS = 58°, angle PSR = 74° and angle QPN = 27°. (i) Find angle PRS. Angle PRS = ................................................ [1] (ii) Find angle PQR. Angle PQR = ................................................ [1] (iii) Find angle RPQ. Angle RPQ = ................................................ [2] © UCLES 2023 0580/43/M/J/23 7 (c) N C B NOT TO SCALE O 34° M A T A, B and C lie on a circle, centre O, with diameter AC. TAM and TBN are tangents to the circle and angle ATO = 34°. Using values and geometrical reasons, complete these statements to show that CB is parallel to OT. In triangles AOT and BOT, OT is common. Angle OAT = angle OBT = 90° because .................................................................................... ..................................................................................................................................................... AT = BT because ........................................................................................................................ ..................................................................................................................................................... Triangle AOT is congruent to triangle BOT because of congruence criterion ........................... Angle AOT = angle BOT = 56° because angles in a triangle add up to 180°. Angle BOC = ...................... ° because ...................................................................................... Angle OBC = ...................... ° because ...................................................................................... ..................................................................................................................................................... CB is parallel to OT because ...................................................................................................... [6] © UCLES 2023 0580/43/M/J/23 [Turn over 8