KCSE 2025 Maths P2 Q5 — Circle: Tangent, Chord & Angles

KCSE 2025 Form 4 Geometry

Published

The Question

“A, B, C and D are points on the circumference of a circle with centre O. The line FE is a tangent to the circle at point C. Angle DCE = 30 degrees and angle BAD = 70 degrees. Giving reasons at each stage, determine the size of the acute angle BOC.”

1

Apply the alternate segment theorem

The tangent CE and the chord CD meet at C making an angle of 30 degrees. By the alternate segment theorem, the angle between a tangent and a chord equals the inscribed angle standing on that chord in the alternate segment. So the angle DAC, subtended by chord CD at the circumference, is also 30 degrees.

∠DAC=∠DCE=30∘(alternate segment theorem)\angle DAC = \angle DCE = 30^\circ \quad \text{(alternate segment theorem)}
2

Split angle BAD to find angle BAC

The chord AC divides the given angle BAD into two parts: angle BAC and angle DAC. Since the whole angle BAD is 70 degrees and the part DAC is 30 degrees, the remaining part BAC is the difference.

∠BAC=∠BAD−∠DAC\angle BAC = \angle BAD - \angle DAC
∠BAC=70∘−30∘=40∘\angle BAC = 70^\circ - 30^\circ = 40^\circ
3

Use the angle at the centre

Angle BAC at the circumference and angle BOC at the centre both stand on the same arc BC. The angle subtended at the centre is twice the angle subtended at the circumference, so doubling angle BAC gives angle BOC.

∠BOC=2×∠BAC(angle at centre = 2 × angle at circumference)\angle BOC = 2 \times \angle BAC \quad \text{(angle at centre = 2 } \times \text{ angle at circumference)}
∠BOC=2×40∘=80∘\angle BOC = 2 \times 40^\circ = 80^\circ

Final Result

The acute angle BOC is 80 degrees.

Why this method works

The alternate segment theorem holds because the tangent is the limiting position of a chord through C, so the tangent-chord angle equals the inscribed angle on the same chord. The angle-at-centre rule follows from isosceles triangles formed by the radii: the central angle always doubles the inscribed angle on the same arc. Chaining these two circle theorems, plus simple angle subtraction at A, links the given 30 and 70 degree angles to the required central angle BOC.

✓

Working back: angle BOC = 80 degrees gives angle BAC = 40 degrees at the circumference, and 40 + 30 (angle DAC) = 70 degrees = angle BAD, which matches the given data.