Theory question 111
Circle Theorems(a) In the diagram, MNPQ is a circle with centre O, |MN| = |NP| and ∠OMN = 50°. Find: (i) ∠MNP; (ii) ∠POQ.
(b) Find the equation of the line which has the same gradient as 8y + 4x = 24 and passes through the point (−8, 12).
Show the model answer
Model answer
(a)(i) MQ is a diameter, so ∠MPQ = 90°. In cyclic quadrilateral MNPQ, ∠NPQ = 180° − 50° = 130°, so ∠MPN = 130° − 90° = 40°
MN = NP, so ∠NMP = 40° and ∠MNP = 180° − 80° = 100°
(ii) ∠PMQ = 50° − 40° = 10°; angle at centre ∠POQ = 2 × 10° = 20°
(b) 8y = −4x + 24 → gradient = −½
y − 12 = −½(x + 8) → 2y − 24 = −x − 8
x + 2y − 16 = 0 (or y = −½x + 8)
Explanation
Use the angle in a semicircle and opposite angles of a cyclic quadrilateral.