Circle Theorems: Tangents & Chords

This note covers key circle theorems related to tangents and chords, including their properties and applications in geometry problems.

Tangent to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact.

Key property: The tangent is perpendicular to the radius drawn to the point of contact.

This means if a radius $\overline{OA}$ meets the circle at point $A$, and $\overline{AT}$ is the tangent at $A$, then:

$\angle OAT = 90^\circ$

There is only one tangent line at any point on a circle.

If two tangents are drawn from an external point to a circle, the lengths of the two tangents from that point to the points of contact are equal.

For example, if $P$ is outside the circle and tangents $PA$ and $PB$ touch the circle at $A$ and $B$ respectively, then:

$\overline{PA} = \overline{PB}$

This property is useful for solving problems involving lengths of tangents.

For instance, if the distance from $P$ to the centre $O$ is $13\,\mathrm{cm}$, and the radius $\overline{OA}$ is $5\,\mathrm{cm}$, then the length of the tangent $\overline{PA}$ can be found using Pythagoras’ theorem in triangle $OAP$:

$$PA = \sqrt{PO^2 - OA^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12\,\mathrm{cm}$$

Chord Properties

A chord is a straight line joining two points on a circle.

Equal chords are equidistant from the centre: If two chords have the same length, their perpendicular distances from the centre are equal.

Conversely, chords that are the same distance from the centre are equal in length.

The perpendicular bisector of any chord passes through the centre of the circle.

This means if you draw a line at right angles to a chord and through its midpoint, this line will always go through the centre.

The relationship between chord length and radius is important:

• Longer chords lie closer to the centre.
• The longest chord in a circle is the diameter, which passes through the centre.

For example, if a chord $AB$ is $10\,\mathrm{cm}$ long and the radius $OA$ is $7\,\mathrm{cm}$, the perpendicular distance $OM$ from the centre $O$ to the chord $AB$ can be found by splitting the chord into two equal parts at $M$ (midpoint):

$$AM = \frac{10}{2} = 5\,\mathrm{cm}$$

Using Pythagoras’ theorem in triangle $OMA$:

$$OM = \sqrt{OA^2 - AM^2} = \sqrt{7^2 - 5^2} = \sqrt{49 - 25} = \sqrt{24} \approx 4.9\,\mathrm{cm}$$

Tangent-Chord Theorem

The tangent-chord theorem states:

The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment of the circle.

This means if a tangent touches the circle at point $A$, and $\overline{AB}$ is a chord, then the angle between the tangent and chord at $A$ equals the angle subtended by chord $AB$ in the opposite segment of the circle.

This theorem is useful for calculating unknown angles in problems involving tangents and chords.

For example, if the angle between the tangent at $A$ and chord $AB$ is $40^\circ$, then the angle subtended by chord $AB$ in the alternate segment of the circle is also $40^\circ$.