Theorems-With-Chords-And-Tangents Wjec-Eduqas Higher

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Exam code:C300

Circles & chords

What is a chord?

A chord is any straight line is a circle that joins any two points on the circumference
- Chords of equal length are equidistant (the same distance) from the centre

Circle Theorem: The perpendicular bisector of a chord passes through the centre

If a line through the centre (such as a radius or diameter) goes through the midpoint of chord
- it will bisect (cut in half) that chord at right angles to it

A circle with a radius bisecting a chord.

To spot this circle theorem on a diagram
- look for a radius and see if it intersects any chords
- or look to see if you could draw a radius that bisects a chord
When explaining this theorem in an exam you can use either phrase below:
- A radius bisects a chord at right angles
- The perpendicular bisector of a chord passes through the centre

Examiner Tips and Tricks

Look out for isosceles triangles formed by a chord and two radii
- Two angles in the triangle will be equal and there will be at least one line of symmetry

Worked Example

The diagram below shows a circle with centre, O.
Two points, P and Q, lie on its circumference.
The radius of the circle is 6 cm.
Angle OPQ = 40º.

Find the length PQ.

A circle of centre, O, has two points on its circumference, P and Q. The lines OQ and PQ are drawn on. The angle OQP = 40º.

Label the radius on the diagram 6 cm

Draw a line from O to the midpoint, M, of the line PQ
The angle formed between the OM and PQ will be a right angle

A diagram showing the same circle with a straight line from the centre to the midpoint of PQ, bisecting it at a right angle.

Use SOHCAHTOA on triangle OMQ to find the length MQ

$table row cell cos space 40 end cell equals cell fraction numerator M Q over denominator 6 end fraction end cell row cell 6 space cos space 40 space end cell equals cell space M Q end cell row cell M Q end cell equals cell 4.59626... end cell end table$

Double MQ to find the length PQ

<img alt=”4.59626… cross times 2 equals 9.19253…” data-mathml=”<math ><semantics><mrow><mn>4</mn><mo>.</mo><mn>59626</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>×</mo><mn>2</mn><mo>=</mo><mn>9</mn><mo>.</mo><mn>19253</mn><mo>.</mo><mo>.</mo><mo>.</mo></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true,”toolbar”:”<toolbar ref=’general’><tab ref=’general’><removeItem ref=’setColor’/><removeItem ref=’bold’/><removeItem ref=’italic’/><removeItem ref=’autoItalic’/><removeItem ref=’setUnicode’/><removeItem ref=’mtext’ /><removeItem ref=’rtl’/><removeItem ref=’forceLigature’/><removeItem ref=’setFontFamily’ /><removeItem ref=’setFontSize’/></tab></toolbar>”}</annotation>

Maths Gcse Wjec-Eduqas Higher