Circles

Circles

Circle questions are theorem-recognition in disguise. Learn the handful below and most CAT circle problems become one-liners.

Essentials

• Area $=\pi r^2$, circumference $=2\pi r$.
• Inscribed-angle theorem: the angle at the centre is twice the angle at the circumference on the same arc; an angle in a semicircle is $90°$.
• Tangent ⊥ radius at the point of contact; the two tangents from an external point are equal.
• Cyclic quadrilateral: opposite angles sum to $180°$.
• Power of a point: for chords through $P$, $PA\cdot PB=PC\cdot PD$.

A worked example

$AB$ is a diameter and $C$ is any other point on the circle. If $\angle CAB=35°$, find $\angle ABC$.

Because $AB$ is a diameter, $\angle ACB=90°$ (angle in a semicircle). The angles of $\triangle ABC$ sum to $180°$, so

$\angle ABC=180°-90°-35°=\mathbf{55°}.$

Common traps

• Central vs inscribed angle. The central angle is double — mixing them up halves or doubles your answer.
• Tangent length. Use the radius-perpendicularity to form a right triangle; the tangent is a leg, not the hypotenuse.
• Arc vs chord. Equal chords subtend equal arcs, but the angle depends on where it's measured.

Checklist

• Look for a diameter → right angle on the circle
• Central angle $=2\times$ inscribed angle
• Tangent ⊥ radius; equal tangents from a point
• Opposite angles of a cyclic quadrilateral sum to $180°$