Circle theorems with algebra

Circle theorems

(1MA1) · Higher

ORIGINAL

Link same-segment angles, a diameter, and a triangle angle sum in one algebraic chain.

Verified against Edexcel 1MA1 (2026 spec)

Question

Points $A$ , $B$ , $C$ and $D$ lie on a circle. $AB$ is a diameter, and $A$ and $D$ lie in the same segment cut off by chord $BC$ . Angle $BAC=(3x+4)^\circ$ and angle $BDC=(5x-10)^\circ$ . Find $x$ and angle $ABC$ .

Every step worked, with the reasoning.

1
$(3x+4)^\circ=(5x-10)^\circ$
Angles $BAC$ and $BDC$ subtend the same chord $BC$ in the same segment, so they are equal.
2
$14=2x$ , so $x=7$
Rearrange and solve the linear equation.
3
$\angle BAC=3(7)+4=25^\circ$
Substitute $x=7$ into either given angle.
4
$\angle ACB=90^\circ$
The angle subtended by diameter $AB$ at the circumference is a right angle.
5
$\angle ABC=180^\circ-90^\circ-25^\circ=65^\circ$
Angles in triangle $ABC$ sum to $180^\circ$ .

Answer: $x=7$ and $\angle ABC=65^\circ$ .

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