GCE Mathematics 2022 Theory — Question 4

Model answer

(a) BCDE is a cyclic quadrilateral, so opposite angles are supplementary: angle BCD + angle BED = 180: (2x+40) + (2y+10) = 180, i.e. 2x+2y+50=180, so x+y=65 ....(i) Angle BAD is the angle at the centre A, and angle BED is the angle at the circumference standing on the same arc BD, so angle BAD = 2 x angle BED (angle at centre = twice angle at circumference): (5x-35) = 2(2y+10) = 4y+20, so 5x - 4y = 55 ....(ii) From (i): y = 65-x. Substitute into (ii): 5x - 4(65-x) = 55, so 5x - 260 + 4x = 55, giving 9x = 315, so x = 35. Then y = 65 - 35 = 30. (b) Since BCDE is a cyclic quadrilateral, angle ABC + angle ADC = 180 (opposite angles of a cyclic quadrilateral are supplementary): angle ABC = 180 - 40 = 140 degrees.