We have looked at some different rules for finding angles and lengths in circles. In this chapter, we are going to recap those rules and look at how to combine them to find values.
Here are the geometrical proofs above written the way you will find them when you have to select them doing your exercises:
Angles at a centre theorem: angle at the centre of a circle is twice the angle at its circumference.
Angles subtended by the same arc theorem: angles at the circumference standing on the same arc are equal.
Angle is a semicircle: the angle in a semicircle is a right angle.
Angles in cyclic quadrilateral: opposite angles in a cyclic quadrilateral are supplementary.
We can use all our existing mathematical knowledge, including the properties of triangles and quadrilaterals, Pythagoras' theorem, as well as congruency proofs to solve problems involving circles and angles.
Here are some additional facts:
where $r$r is the radius of the circle $c$c is the angle subtended at the centre by the chord or $\text{Chord length }=\sqrt{r^2-d^2}$Chord length =√r2−d2, where $r$r is the radius of the circle and $d$d is the perpendicular distance from the chord to the centre of the circle.
Here are the Mathspace geometrical reasonings you may need when calculating distances in circles, exactly as you will find them in the exercises:
Let's use this info to find some missing values.
Calculate $x$x and give reasons for your answer.
In the diagram, $\angle BAC=45^\circ$∠BAC=45°. Calculate $x$x and give reasons for your answer.
Calculate the angle marked $x$x at the centre of the circle. Give reasons for your answer related to the properties of a circle.