Circle Theorems 3
Maths: Edexcel IGCSE Higher, topic 4.5.4
Circle Theorems
Angles in the same segment are equal
- All triangles drawn from a chord in the same segment will have the same angle at the circumference.
Proof
- Draw in radii from the ends of the chord.
- Label the angle at the centre.
Proof continued
- Using ‘angle at centre twice that at the circumference’ we see that
z = 2x, z = 2y and so x = y.
Circle Theorems
Alternate segment theorem
- The angle between a tangent and a chord is always equal to the angle at the circumference of any triangle formed from the chord.
Proof
- Draw in a diameter from the point where the tangent touches. Connect it to the other end of the chord.
Proof continued
- Using ‘radius and tangent meet at a right angle’ we get a right angle where the diameter meets the tangent.
- Using ‘angle in a semicircle is 90°’ we get that the angle at the other end of the chord is a right angle too.
Proof continued
- We find that the angle at the bottom of the diameter is the same as x.
- Using ‘angles in same segment’ we get that x = y.
Circle Theorems
Internal intersecting chord theorem
- If two chords intersect inside the circle, the following relationship is true:
- |AS| × |SB| = |CS| × |SD|
External intersecting chord theorem
- If two chords intersect outside of the circle, the following relationship is true:
- |MN| × |NO| = |OP| × |PQ|
Test yourself on Circle Theorems 3
What is angle x?
What is angle x?
What is angle x?
What is angle x?
What is angle x in the diagram?