Circle Theorems (Edexcel IGCSE Maths A)

True.

This is the circle theorem: The perpendicular bisector of a chord passes through the centre.

If a triangle is formed from two radii and a chord within a circle it will be an isosceles triangle as the two radii are two sides of the same length.

The circle theorem is: A radius and a tangent are perpendicular.

The distances are equal. This is the circle theorem: Tangents from an external point are equal in length.

A kite is formed by the centre of a circle, two tangents to the circle that intersect, and two radii

The two radii form two adjacent sides of equal length and the tangents form the other pair of adjacent sides of equal length. The angle between each radius and tangent is 90º.

True.

For two chords, AB and CD that meet at point P the ratio of longer lengths ≡ ratio of shorter lengths (of chords). This is known as the intersecting chord theorem.

The easiest way to use the intersecting chord theorem is to use it in its multiplicative form, AP × PB = CP × PD.

A secant is a line that extends through a circle, intersecting it at two points.

True.

The intersecting secant theorem is a version of the intersecting chords theorem and applies when two chords intersect outside the circle instead of inside.

For two intersecting secants, ABP and CDP that meet at point P, the easiest way to use the intersecting secant theorem in the form BP(AB + BP) = DP(CD + DP)

A cyclic quadrilateral is a quadrilateral inside a circle with all four vertices of the quadrilateral lying on the circumference of the circle.

Circle theorem: Opposite angles in a cyclic quadrilateral add up to 180º.

Angle ABC + Angle ADC = Angle BAD + Angle BCD = 180º.

False.

For a quadrilateral in a circle that has one vertex at the centre of the circle and the other three on the circumference, opposite angles do not add up to 180º.

All four vertices must lie on the circumference of the circle for it to be a cyclic quadrilateral.

A segment is a region bounded by a chord and an arc of a circle.

The circle theorem in the diagram is: Angles at the circumference subtended by the same arc are equal or Angles in the same segment are equal.

A cyclic triangle is a triangle within a circle where all three vertices of the triangle lie on the circumference of the circle.

The circle theorem in the diagram is: The alternate segment theorem.

The angle between a chord and a tangent is equal to the angle in the alternate segment.

False.

Identifying a cyclic triangle can help you to spot the alternate angle theorem, but it is not enough on its own.

You must also look to make sure that one of the vertices of the cyclic triangle touches the same point on the circumference as a tangent to the circle.