GCE Mathematics 2022 Theory — Question 11

Model answer

(a) The exterior angles of any polygon sum to 360 degrees: 42+38+57+x+(x+y)+(2x-15)+(3x-y) = 360. The y-terms cancel (+y and -y): 42+38+57-15 + 7x = 360, i.e. 122 + 7x = 360. 7x = 238, so x = 34. Since x is 7 degrees less than y: y = x+7 = 41. (b) Since OX=OZ (radii), triangle OXZ is isosceles: angle OZX = angle OXZ = 34 degrees. Angle XOZ = 180 - 34 - 34 = 112 degrees (angle sum of a triangle). Angles at the point O around the circle sum to 360 degrees. If angle XOZ (112) and angle XOY (146) are the two angles on either side of Z relative to Y going around O, then angle ZOY = 360 - 112 - 146 = 102 degrees. Since OZ=OY (radii), triangle OZY is isosceles: angle OZY = angle OYZ = (180-102)/2 = 39 degrees. Therefore angle OYZ = 39 degrees.