Theory question 105
Circle Theorems(a) In the diagram, A, B, C and D are points on the circumference of a circle. XY is a tangent at A. Find: (i) ∠CAX; (ii) ∠ABY
(b) If (m + 1) and (m − 3) are factors of m2 − km + c, find the values of k and c.
Show the model answer
Model answer
(a)(i) ∠ACB = ∠ADB = 20° (same segment). C, B and Y are in a straight line, so in triangle ACY the exterior angle ∠CAX = ∠ACY + ∠AYC = 20° + 60° = 80°.
(ii) ∠BAY = ∠ADB = 20° (angle between tangent and chord equals the angle in the alternate segment). In triangle ABY, ∠ABY = 180 − 60 − 20 = 100°.
(b) (m + 1)(m − 3) = m2 − 2m − 3, so k = 2 and c = −3.
Explanation
Use the tangent–chord angle and the exterior angle of a triangle.