Theory question 104
Circle TheoremsIn the diagram, MNPQ is a circle with centre O. ∠MNQ = y°, ∠NEP = x°, ∠NMP = (100 − x)° and ∠NFP = (5x)°. Find the values of x and y.
Show the model answer
Model answer
∠NQP = ∠NMP = 100 − x (same segment), so arc NP subtends 2(100 − x) at the centre
Angle between secants: ∠NEP = ½(arc NP − arc MQ) ⇒ arc MQ = 200 − 2x − 2x = 200 − 4x
Angle between chords: ∠NFP = ½(arc NP + arc MQ) = ½(400 − 6x) = 200 − 3x = 5x ⇒ x = 25
y = ∠MNQ = ½ arc MQ = 100 − 2(25) = 50
Explanation
Angles in the same segment are equal; use the angle sum in the triangles at F and E.