Chapter 21: Circle geometry and circle theorems

The chapter problem

A circular window has a straight line across it, from one edge to another. This chord is $13$ cm long, and the radius is $7$ cm. Predict whether the chord could fit in the window. Give one reason for your prediction.

Two points on the circumference lie in the same segment of the circle. They see the chord at angles of $38^\circ$ and $x^\circ$. A radius from the centre meets the chord at right angles. Predict a possible value of $x$ and one fact that could help you. You do not need the answer yet.

By the end of this chapter you will be able to

• name and use the centre, radius, chord, diameter, circumference, tangent, arc, sector and segment;

• use circle angle theorems to calculate missing angles;
• use tangent and chord properties to calculate lengths and angles;
• construct clear proofs that use circle theorems and known angle facts.

Route through the chapter

1. 21.1 Parts of a circle introduces the language and parts of a circle.

1. 21.2 Circle theorems uses angles at the centre and circumference, semicircles, segments and cyclic quadrilaterals.
2. 21.3 Tangents and chords applies tangent and chord properties, including the alternate segment theorem.
3. 21.4 Proof of circle theorems builds connected reasons into a proof.

At the marked Progress checks, return to the window problem and use the new facts you have learned.

Make a connection

Circles appear in a wheel, a round window, a clock, and many other designs. Choose one example that matters to you or someone you know. Write one sentence explaining where a circle fact from this chapter could help.