Chords & Tangents (CIE IGCSE Maths: Extended)

What is a chord?

• A chord is any straight line is a circle that joins any two parts of the circumference
• Any equal chords are equidistant from the centre of the circle
  • This means that if each end of a chord is joined to two radii, they will form an isosceles triangle
  • 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 chord 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
  • It is often seen in questions involving congruent triangles
• 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 

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
• This circle theorem states that a radius and a tangent meet at 90°
  • Perpendicular just means at right angles
• When using this theorem in an exam you must use the keywords 
  • 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
• 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
  • In this sense the tangents end at two points – the first point is where the two tangents meet and the other end is where each one touches the circle