Tangents & Circle Properties

Three classical circle facts, translated into coordinates, power the harder questions in this topic. Each converts a geometric word in the question into a gradient or midpoint computation you already know.

Fact 1. Tangent $\perp$ radius at the point of contact

The workhorse. To find the tangent at $P$ on a circle with centre $C$:

Circle $(x - 2)^2 + (y - 3)^2 = 5$, tangent at $P(1, 5)$. Gradient of radius $CP$: $\frac{5 - 3}{1 - 2} = -2$ (M) Tangent gradient $= \frac{1}{2}$, negative reciprocal, with “tangent $\perp$ radius” stated (M) Tangent: $y - 5 = \frac{1}{2}(x - 1) \to$ $2y = x + 9$ (A)

Three steps, three marks, no calculus needed. The normal at $P$ is the radius line $CP$ itself, extended both ways.

Fact 2. The centre lies on the perpendicular bisector of any chord

Equivalently: the perpendicular from the centre to a chord bisects it. Uses:

• Find the centre from two chords: two perpendicular bisectors, intersect, done
• Chord midpoint problems: the line from centre to chord-midpoint is perpendicular to the chord, a hidden $m_1 m_2 = -1$ condition
• Half-chord Pythagoras: $(\text{half chord})^2 + (\text{centre-to-chord distance})^2 = r^2$, the slick route to chord lengths without finding endpoints

Fact 3. The angle in a semicircle is $90^\circ$

If $AB$ is a diameter, any point $P$ on the circle has angle $APB = 90^\circ$. Coordinate translations: “show $AB$ is a diameter” $\to$ show the midpoint of $AB$ is the centre (and $|AB| = 2r$); “show $P$ lies on the circle with diameter $AB$” $\to$ show gradients $PA$ and $PB$ multiply to $-1$. The conclusion sentence, as ever, carries its own mark.

Choosing the method for “show it’s a tangent”

Two accepted routes: discriminant $= 0$ after substitution, or perpendicular distance from centre $=$ radius. Pick the one you’ve drilled (the discriminant reuses standard machinery); state which property you’re invoking either way.

Common mistakes

• Tangent built through the centre instead of the contact point
• Radius gradient used directly for the tangent (forgot the reciprocal flip)
• The geometric fact applied silently, name it (“tangent $\perp$ radius”), it’s creditable
• Half-chord Pythagoras with the full chord
• “Diameter” claims missing one of the two requirements (midpoint $=$ centre AND on the circle)

Full topic context: Circle Geometry notes, and remember this whole topic is new for 2025-2027, so practise from specimen and recent papers.