Tangents to a circle

5 minAS Level1.3.4

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find the equation of the tangent to a circle at a given point, using the fact that the tangent is perpendicular to the radius.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$C$	centre of the circle
$P$	point of contact (where the tangent touches)
$r$	radius
$m$	gradient of the tangent
$(x_1, y_1)$	a known point on the line (here $P$ )

The one fact you need

Spin a weight on a string, then let go. It flies off at right angles to the string, along the tangent.

Where the analogy breaks: real physics adds gravity and wobble. The geometry is the point.

For a circle, that's exact. Let $C$ be the centre and $P$ a point on the circle.

The tangent at $P$ touches the circle only at $P$ .
It is perpendicular to the radius $CP$ (meets it at a right angle).

Definition

Tangent to a circle

A tangent to a circle is a straight line that meets the circle at exactly one point, the point of contact. At that point, the tangent is perpendicular to the radius.

The four-step recipe

Centre. Get it from the equation (complete the square if it's expanded).
Radius gradient. Gradient of $CP$ .
Flip and change the sign. Tangent gradient $= -\dfrac{1}{\text{radius gradient}}$ .
Line through $P$ . $y - y_1 = m(x - x_1)$ .

Optional step 0: check that $P$ really is on the circle by substituting it.

Quick check

Fill in the blanksQuestion 1 · Recall

CalculateQuestion 2 · Recall

Answer the 2 questions above to continue.