Circle facts that find circles

8 minAS Level1.3.4

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

By the end of this lesson

By the end you can use the angle in a semicircle and the perpendicular bisector of a chord to find the centre and equation of a circle through given points.

Words and notation first

A chord is a straight line joining two points on a circle. A diameter is a chord through the centre.
A tangent is a line that touches the circle at exactly one point.
Perpendicular ( $\perp$ ) means at right angles. The perpendicular bisector of $AB$ is the line that cuts $AB$ in half and is at right angles to it.
$m_{AP}$ means "the gradient of the line from $A$ to $P$ ". The letters underneath just name the line.

The key coordinate fact: two lines are perpendicular exactly when their gradients multiply to $-1$ . For example, gradients $2$ and $-\tfrac12$ : $2 \times \left(-\tfrac12\right) = -1$ , so lines with these gradients meet at right angles.

Three facts, endless questions

Circle facts you may have met before, now with coordinates. Three of them do almost all the work:

Fact	In coordinates
Tangent ⊥ radius	gradients multiply to $-1$
Angle in a semicircle is $90°$	if $AB$ is a diameter, $m_{AP} \times m_{BP} = -1$
Perpendicular bisector of a chord passes through the centre	the centre lies on it

Definition

Angle in a semicircle

If $AB$ is a diameter of a circle and $P$ is any other point on the circle, then angle $APB = 90°$ .

It works backwards too: if angle $APB = 90°$ , then $P$ lies on the circle with diameter $AB$ .

Why is it $90°$ ? Join $P$ to the centre $O$ . Now $OA$ , $OB$ and $OP$ are all radii, so triangles $AOP$ and $BOP$ are both isosceles. Call their base angles $x$ (at $A$ and at $P$ ) and $y$ (at $B$ and at $P$ ). The big triangle $APB$ has angles $x$ , $y$ and $x + y$ , which add to $180°$ : $2x + 2y = 180°$ , so the angle at $P$ is $x + y = 90°$ .

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.