O lvl E Maths: Properties of Circles

RI Sec 3 EOY 1998 Q11

In the diagram above, DE is the tangent at D to the circle with centre O. AC is the diameter, ∡ACD = 30° and ∡BAC = 67°. Calculate
(a) ∡CAD,
(b) ∡CDE,
(c) ∡DBO.

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Answer:

(a) ∡CDA = 90° (angle at semi-circle)
∡CAD = 180° - 90° - 30° = 60° (angles in a triangle)

(b) ∡CDE = ∡CAD = 60° (alternate segment)

(c) ∡ABD = ∡ACD = 30°
∡ABO = ∡OAB = 67° (angles in an isosceles triangle)
∡DBO = ∡ABO - ∡ABD
∡DBO = 67° - 30° = 37°

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