How do you show in coordinates that angle APB is 90^\circ?

Show that the gradients of AP and BP multiply to -1.

State the chord property.

The perpendicular from the centre to a chord bisects the chord.

State the tangent property.

A tangent is perpendicular to the radius at the point of contact.

Centre and radius of a circle with diameter AB?

Centre is the midpoint of AB; radius is \frac12AB.

Midpoint of (x1,y1) and (x2,y2)?

\left(\frac{x1+x2}{2},\frac{y1+y2}{2}\right)

Tangent at (4,0) to a circle with centre (1,0)?

The radius is horizontal, so the tangent is vertical: x=4.

How do you find where a line meets a circle?

Substitute the line into the circle equation and solve the quadratic.

Circle propertiesEdexcel International A Level Maths: Flashcards

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State the angle in a semicircle property.

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State the angle in a semicircle property.: The angle subtended at the circumference by a diameter is $90^\circ$ .
How do you show in coordinates that angle $APB$ is $90^\circ$ ?: Show that the gradients of $AP$ and $BP$ multiply to $-1$ .
State the chord property.: The perpendicular from the centre to a chord bisects the chord.
State the tangent property.: A tangent is perpendicular to the radius at the point of contact.
Equation of a circle with centre $(a,b)$ and radius $r$ ?: $(x-a)^2+(y-b)^2=r^2$
Condition for two gradients to be perpendicular?: $m_1m_2=-1$
Gradient of a tangent if the radius has gradient $\frac43$ ?: $-\frac34$
Centre and radius of a circle with diameter $AB$ ?: Centre is the midpoint of $AB$ ; radius is $\frac12AB$ .
Distance from the centre to a chord of half-length $h$ in a circle of radius $r$ ?: $\sqrt{r^2-h^2}$
Midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ ?: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
Tangent at $(4,0)$ to a circle with centre $(1,0)$ ?: The radius is horizontal, so the tangent is vertical: $x=4$ .
How do you find where a line meets a circle?: Substitute the line into the circle equation and solve the quadratic.

Exam questions on Circle properties

A circle has diameter $AB$ , where $A=(1,2)$ and $B=(9,8)$ . The point $P=(8,1)$ lies on the circle.
Show that angle $APB$ is a right angle, and name the circle property you are using.2 marks
A circle has centre $C=(2,3)$ and radius $5$ . The points $P=(5,7)$ and $Q=(6,0)$ lie on the circle.
Find the exact distance from $C$ to the chord $PQ$ .2 marks
The circle $S$ has equation $(x-1)^2+(y+2)^2=25$ . The point $T=(4,2)$ lies on $S$ .
Find an equation of the tangent to $S$ at $T$ , giving your answer in the form $ax+by+c=0$ where $a$ , $b$ and $c$ are integers.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).