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Chords & Tangents (Edexcel IGCSE Maths)
Revision Note
Circles & Chords
What is a chord?
- A chord is any straight line is a circle that joins any two parts of the circumference
- An isosceles triangle is formed by a chord and the two radii joining the ends of the chord to the centre of the circle
- This is not technically a circle theorem, but is very useful in answering circle theorem questions
- To start any circle theorem questions
- first identify any radii and mark them as equal lines
- then look to see if the radii are joined to any chords
Circle Theorem: The perpendicular bisector of a chord is a radius
- If a radius or diameter intersects a chord in a circle, in will bisect that short at a right angle
- bisect means to cut in half
- This circle theorem is seen less often, but can be very useful in finding equal lengths and angles
- To spot it, look for a radius and see if it intersects any chords
- Problems involving this theorem often have the radii being joined to the end of the chords and so creating two congruent triangles
- This is also easier to see than remember from its description
What else should I know about chords?
- Although it is not strictly a circle theorem the following is a very important fact for solving some problems
- A triangle which is formed from the centre using a chord and two radii is an isosceles triangle
- This means at least two of the angles will be equal and there will be at least one line of symmetry
- This is very useful in proving circle theorems
Circles & Tangents
What is a tangent?
- A tangent to a circle is a straight line outside of the circle that touches its circumference only once
- Tangents are the easiest thing to spot quickly in a circle theorem question as they lie outside of the circle and stand out clearly
Circle Theorem: A radius and a tangent are perpendicular
- Most of the time, if there is a tangent in a circle theorem question it will meet a radius at the point where it touches the circumference of a circle
- Make sure that the line the tangent meets is definitely a radius; that it starts at O, the centre
- This circle theorem states that a radius and a tangent meet at 90°
- Perpendicular just means at right angles
- If asked to state reasons and you use this theorem then use the key phrase;
- "A radius and a tangent meet at right angles"
What else should I know about tangents?
- Although it is not strictly a circle theorem the following is a very important fact for solving some problems
- Two tangents from a circle to the same point outside of a circle are equal length
- This means that a kite can be formed by two tangents meeting a circle
- Remember that a kite is essentially two congruent triangles about its main diagonal
- The kite will have two right angles, where the tangents meets the radii
Examiner Tip
- If you spot a tangent on a circle diagram, look to see if it meets a radius and add in the right angle clearly to the diagram straight away
- In some cases just the act of doing this can earn you a mark!
Worked example
Find the value of θ in the diagram below.
The lines ST and RT are both tangents to the circle and meet the two radii on the circumference at the points S and T.
Angle TSO = angle TRO = 90° (A radius and a tangent meet at right angles)
Use vertically opposite angles to find the value of the angle at T that is opposite the 25° angle.
Angle RTS = 25° (vertically opposite angles)
Mark these angles clearly on the diagram.
Angles in a quadrilateral add up to 360°. Use this to form an equation for θ.
Simplify.
Solve.
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