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Applications of Circle Geometry

Lesson

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.

 

Angles in Circles

  • Angle at the centre theorem: the angle at the centre is twice the size of the inscribed angle.
  • Angles subtended by the same arc theorem: when there are two fixed endpoints, the angle is always the same, no matter where it is on the circumference.
  • Angle in a semicircle: The angle inscribed in a right angle is always a right angle ($90^\circ$90°).
  • Angles in cyclic quadrilaterals: The opposite angles in a cyclic quadrilateral add up to $180^\circ$180°.
Mathspace Talk

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.

 

Distances in Circles

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:

  • The radii in a circle are always equal.
  • To calculate a chord length: $\text{Chord length }=2r\sin\left(\frac{c}{2}\right)$Chord length =2rsin(c2)  

     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 =r2d2, 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.

  • The perpendicular line from the centre of a circle to a chord bisects that chord.
Mathspace Talk

Here are the Mathspace geometrical reasonings you may need when calculating distances in circles, e