Circle Theorems (Edexcel IGCSE Maths A (Modular))
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What is an arc?
An arc is a portion of the circumference of a circle.
What is the circle theorem shown below?
The circle theorem shown in the diagram is: The angle subtended by an arc at the centre is twice the angle at the circumference.
Note that both angles need to share the same arc (it should look like an arrow head not a kite).
True or False?
False.
For the angle at the centre to be twice that of the angle at the circumference, both angles must be subtended by the same arc.
What is the circle theorem shown below?
The circle theorem in the diagram is: The angle in a semicircle is a right angle.
If one edge of the triangle inside a circle is the diameter, then the triangle is contained within half of the circle (a semicircle) and the angle opposite the diameter is 90º.
True or False?
A triangle in a circle, where all three points touch the circumference of the circle, will always be a right-angled triangle.
False.
A triangle in a circle, where all three points touch the circumference of the circle, will not always be a right-angled triangle.
It will only be a right-angled triangle if one of the sides is the diameter of the circle.
True or False?
The angle in a semicircle is a right angle circle theorem is a special case of the angle subtended by an arc at the centre is twice the angle at the circumference circle theorem.
True.
The angle in a semicircle is a right angle circle theorem is a special case of the angle subtended by an arc at the centre is twice the angle at the circumference circle theorem.
What is a chord?
A chord is any straight line that joins two points on the circumference of a circle.
What is a tangent?
A tangent is a straight line that touches a circle at exactly one point.
What is a radius?
A radius is a line segment connecting the centre of a circle to a point on the circumference.
Define the term perpendicular bisector.
A perpendicular bisector is a line that cuts another line exactly in half (bisects) and crosses it at a right angle (perpendicular).
True or False?
The perpendicular bisector of a chord passes through the centre of the circle.
True.
This is the circle theorem: The perpendicular bisector of a chord passes through the centre.
What type of triangle is formed in a circle from two radii and a chord?
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.
What is the circle theorem that describes the relationship between a radius and a tangent?
The circle theorem is: A radius and a tangent are perpendicular.
If two tangents to a circle intersect, what can be said about the distances on each line between the point of intersection and the circle?
The distances are equal. This is the circle theorem: Tangents from an external point are equal in length.
What geometrical shape is formed by the centre of a circle, two tangents to the same circle that intersect, and two radii?
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 or False?
For two chords, AB and CD that meet at point P, the ratio of longer lengths (of chords) ≡ ratio of shorter lengths (of chords).
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.
How is the intersecting chord theorem used to solve problems?
The easiest way to use the intersecting chord theorem is to use it in its multiplicative form, AP × PB = CP × PD.
What is a secant?
A secant is a line that extends through a circle, intersecting it at two points.
True or False?
The intersecting secant theorem applies when two chords intersect outside the circle.
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.
How is the intersecting secant theorem used to solve problems?
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)
What is a cyclic quadrilateral?
A cyclic quadrilateral is a quadrilateral inside a circle with all four vertices of the quadrilateral lying on the circumference of the circle.
What is the circle theorem that describes the relationship between angles in a cyclic quadrilateral?
Circle theorem: Opposite angles in a cyclic quadrilateral add up to 180º.
Angle ABC + Angle ADC = Angle BAD + Angle BCD = 180º.
True or False?
Opposite angles in the quadrilateral below add up to 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.
What is a segment?
A segment is a region bounded by a chord and an arc of a circle.
Identify the circle theorem in the diagram below.
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.
What is a cyclic triangle?
A cyclic triangle is a triangle within a circle where all three vertices of the triangle lie on the circumference of the circle.
Identify the circle theorem in the diagram below.
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.
True or False?
You can spot the alternate segment theorem by just identifying a cyclic triangle.
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.