Theorems with Chords & Tangents (Cambridge (CIE) IGCSE Maths)

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

• 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

Circles & Tangents

What is a tangent?

• A tangent to a circle is a straight line outside of the circle that touches its circumference at exactly one point

Circle Theorem: A radius and a tangent meet at right angles

• If a radius and a tangent meet at a point on the circumference of a circle, the angle formed between them will be 90°

  • They are perpendicular to each other

• When explaining this theorem in an exam you must use the keywords:

  • A radius and a tangent meet at right angles

Circle Theorem: Tangents from an external point are equal in length

• Two tangents from the same external point are equal in length

• This means that a kite can be formed by two tangents meeting a circle

  • The kite below has a vertical line of symmetry

    • It is formed from two congruent triangles back-to-back

  • The kite will have two right angles where the tangents meet the radii

    • You can use Pythagoras and SOHCAHTOA on each of these triangles