Theory question 112
Circle Theorems(a) In the diagram, P, Q, R and S are points on the circle with centre O. QR//OS, ∠QOR = 2m, ∠QPR = n and ∠SOR = 54°. Find the values of m and n.
(b) The length of a rectangle is 4 cm more than the width. If the perimeter is 40 cm, find the area.
Show the model answer
Model answer
(a) ∠ORQ = ∠SOR = 54° (alternate angles, QR // OS)
OQ = OR, so ∠OQR = 54° and ∠QOR = 180° − 108° = 72° ⇒ 2m = 72°, m = 36°
n = ∠QPR = ½∠QOR = 36° (angle at centre is twice angle at circumference)
(b) 2(w + w + 4) = 40 ⇒ w = 8 cm, length = 12 cm; area = 96 cm²
Explanation
Use alternate angles with the parallel lines, then the isosceles triangle OQR.