Circles

The One Thing

A tangent touches a circle at exactly one point and is perpendicular to the radius drawn to that point of contact. Tangents drawn from the same external point have equal lengths, allowing unknown lengths and angles to be determined through right-triangle geometry and congruence.

Definitions and Results

• Circle: A set of all points in a plane that are at the same distance from a fixed point called the centre.
• Centre: The fixed point from which every point on the circle is at an equal distance.
• Radius: A line segment joining the centre of a circle to any point on the circle.
• Tangent: A line that intersects a circle at exactly one point.
• Point of contact: The single point where a tangent touches the circle.
• Secant: A line that intersects a circle at two distinct points.
• External point: A point lying outside the circle from which tangents may be drawn.
• Radius-tangent theorem: The radius through the point of contact is perpendicular to the tangent. If $PT$ is tangent to a circle with centre $O$ at $T$, then $OT\perp PT$, so $\angle OTP=90^{\circ }$.
• Converse of the radius-tangent theorem: If a line through a point on a circle is perpendicular to the radius at that point, then the line is a tangent to the circle.
• Equal tangents theorem: The lengths of two tangents drawn from an external point to a circle are equal. If $PA$ and $PB$ are tangents from an external point $P$, then $PA=PB$.
• Number of tangents from a point: From a point inside a circle, no tangent can be drawn; from a point on the circle, exactly one tangent can be drawn; from a point outside the circle, exactly two tangents can be drawn.
• Radii to points of contact: If $OA$ and $OB$ are radii to the points of contact $A$ and $B$, then $OA=OB$.
• Distance from centre to tangent: The distance from the centre to a tangent equals the radius.
• Tangent-length equation: If $OP$ is the distance from the centre to an external point and $PT$ is a tangent, then

$$P{T}^2=O{P}^2-O{T}^2,$$ by the Pythagorean theorem.

Worked Methods

Establishing a tangent using the radius-tangent theorem

• Identify the point of contact and the centre of the circle.
• Draw or identify the radius from the centre to the point of contact.
• Show that the radius is perpendicular to the proposed tangent.
• Conclude that the line is tangent to the circle.

For example, if $PT$ is tangent to a circle with centre $O$ at $T$, then $$OT\perp PT$$ and therefore $$\angle OTP=90^{\circ }.$$

Proving that a line is tangent using the converse

• Identify a point on the circle and the radius drawn to that point.
• Show that the proposed line passes through the point on the circle.
• Establish that the line is perpendicular to the radius.
• Apply the converse of the radius-tangent theorem to conclude that the line is tangent.

Proving equal tangents from an external point

• Let $PA$ and $PB$ be tangents from an external point $P$.
• Join the centre $O$ to the points of contact $A$ and $B$, forming $OA$ and $OB$.
• Use the radius-tangent theorem to establish

$$OA\perp PA\text{and}OB\perp PB.$$

• Recognise that triangles $OAP$ and $OBP$ are right triangles.
• Use $OA=OB$, since both are radii, and $OP=OP$, since it is common to both triangles.
• Prove that triangles $OAP$ and $OBP$ are congruent by the right-triangle hypotenuse-side condition.
• Corresponding sides are equal, so

$$PA=PB.$$

Finding a tangent length

• Identify the right triangle formed by the centre, the external point, and the point of contact.
• Use the radius-tangent theorem to establish the right angle.
• Apply the Pythagorean theorem.
• If $OP$ is the distance from the centre to the external point, $OT$ is the radius, and $PT$ is the tangent, use

$$O{P}^2=O{T}^2+P{T}^2.$$

• Rearrange to obtain

$$P{T}^2=O{P}^2-O{T}^2.$$

Determining how a line meets a circle

• Consider the shortest distance from the centre of the circle to the line.
• Compare this distance with the radius.
• If the distance equals the radius, the line is a tangent and has exactly one intersection with the circle.
• If the distance is smaller than the radius, the line has two intersections with the circle.
• If the distance is larger than the radius, the line has no intersections with the circle.

Where It Goes Wrong

• Treating a line that meets the circle at two points as a tangent; such a line is a secant.
• Forgetting that a tangent and a circle have exactly one common point.
• Omitting the condition that the radius must be drawn to the point of contact before applying the radius-tangent theorem.
• Assuming that every point can produce tangents: a point inside the circle produces none, a point on the circle produces exactly one, and an external point produces exactly two.
• Using equal tangent lengths without identifying that both tangents originate from the same external point.
• Applying the Pythagorean theorem without first establishing the right angle from the perpendicular radius and tangent.

What Gets Asked

• Define a circle, centre, radius, tangent, point of contact, secant, or external point.
• State or apply the radius-tangent theorem.
• Use the converse of the radius-tangent theorem to prove that a line is tangent.
• Determine the number of tangents from a point inside, on, or outside a circle.
• Prove that $PA=PB$ for tangents $PA$ and $PB$ drawn from an external point.
• Prove congruence of triangles $OAP$ and $OBP$ using the right-triangle hypotenuse-side condition.
• Calculate a tangent length using

$$P{T}^2=O{P}^2-O{T}^2.$$

• Determine whether a line has zero, one, or two intersections with a circle by comparing its shortest distance from the centre with the radius.
• Use tangent properties to determine unknown lengths, angles, and geometric relationships in a diagram.