Circles — Tangents & Chords

Section 1 of 3

Parts of a Circle

The words

Radius: centre to edge. Diameter: right across through the centre, $D = 2r$. Chord: a line joining two points on the circle. Arc: part of the edge. Sector: a ‘pizza slice’. Tangent: a line touching the circle at one point. The edge length is the circumference.

Section 2 of 3

Tangent Meets Radius at 90°

Always perpendicular

A tangent and the radius drawn to the point of contact always meet at $90^\circ$. That makes a right-angled triangle — use Pythagoras.

Worked example — distance to an external point

A tangent of length $12\,\text{m}$ touches a circle of radius $8\,\text{m}$. Find $x$, the distance from the centre $O$ to the external point.

Radius $\perp$ tangent, so $x$ is the hypotenuse

$x^2 = 8^2 + 12^2 = 64 + 144 = 208$

$x = \sqrt{208} = 4\sqrt{13}$

$x = 4\sqrt{13}$

You try

From an external point $B$, $|OB| = 14$ and the tangent $|AB| = 10$. Find the radius $|OA| = x$.

Radius $\perp$ tangent: $x^2 + 10^2 = 14^2$.

$x^2 = 196 - 100 = 96 \Rightarrow x = \sqrt{96}$

$x = \sqrt{96}$

Section 3 of 3

A Chord Cut in Half

Perpendicular from the centre

A radius that meets a chord at $90^\circ$ bisects it (cuts it exactly in half). Cutting a chord in half also makes a $90^\circ$ angle.

Worked example — length of a chord

A circle has radius $8$. A chord is $3$ from the centre. Find half the chord ($x$), then the full chord $|AB|$.

Right angle at the midpoint: $x^2 + 3^2 = 8^2$

$x^2 = 64 - 9 = 55 \Rightarrow x = \sqrt{55} = 7.4$

$|AB| = 2x = 2\sqrt{55} = 14.8$

$|AB| = 14.8$

That’s Class 7.

Circle parts, tangent $\perp$ radius, and a chord bisected by a perpendicular radius. Class 8: the semicircle and two radii.