Geometry Circle theorems provide the rules and principles needed to understand the relationships between angles and lengths in circles. These theorems are important in GCSE Maths and beyond, applied to various problem-solving scenarios involving circular shapes. Mastering geometric circle theorems will improve your ability to tackle geometry questions effectively.
This guide explains the main theorems. Examples and quizzes are included to test your knowledge. If you need further support, consider looking through TeachTutti's list of qualified Maths tutors for personalised guidance in revising this topic.
Circle theorem 1 - Angle at the centre
Circle theorems describe the relationships between angles and segments in and around circles. These theorems are used to analyse geometric shapes and solve problems that involve circles.
The Angle at the Centre theorem deals with the angles subtended by the same arc at the centre and any point on the circumference. It states that the angle subtended at the circle's centre by two points on the circumference is exactly twice the angle subtended at any point on the circumference in the same segment.
Look at the following example. There are two points (A and B) on the circumference and lines are drawn from these points to another point on the circumference (C) and also the circle's centre (O). This creates two angles. The angle at the centre (AOB) is exactly twice as large as the angle at the circumference (ACB) when both are subtended by the same arc between points A and B. This consistent relationship is what the Angle at the Centre Theorem specifies and it plays a crucial role in proofs and applications within circle geometry.
Example:
You have points A and B on a circle and a third point C on the circumference opposite the center O from A and B. If the angle ACB (at the circumference) measures 30 degrees, the Angle at the Centre Theorem tells us that the angle AOB at the centre must measure 60 degrees.
1
If an angle at the circumference of a circle measures 40° between two points, what is the corresponding angle at the centre formed by lines from these points to the centre?
Circle theorem 2 - Alternate Segment
The Alternate Segment theorem shows the relationships between tangents and chords from a particular point on a circle. This theorem states that the angle between a tangent and a chord through the point of contact is the same as the alternate segment. It's used in geometric proofs and when solving problems with tangents and chords.
Imagine a circle with a tangent line touching it at point A and a chord BC passing through A. If you draw an angle BAC between the tangent at A and the chord AB, this angle is congruent to the angle BCA in the opposite segment of the circle. This relationship holds as long as the chord and tangent maintain the point of contact at A.
Example:
Let's say we have a circle where a tangent (P) touches the circle and a chord (PQ) is drawn through the tangent, extending to meet the circle (Q). If the angle between the tangent and the chord (angle TPQ) measures 50 degrees, the angle in the opposite segment formed by the chord (angle PQT) will also measure 50 degrees.
2
A circle has a tangent AB and a chord AC that meets at point A. If the angle between tangent AB and chord AC is 35 degrees, what is the angle in the alternate segment at point C?
Circle theorem 3 - Perpendicular Chord Bisector
The Perpendicular Chord Bisector theorem states that any perpendicular line from the circle's centre to a chord will bisect this chord. It's used to solve geometric constructions and proofs using circles.
For example, a chord (AB) exists inside a circle and the circle's centre is marked as O. When a line is drawn from the centre (O) perpendicular to the chord (AB), this line will not intersect the chord at its midpoint and bisect it into equal halves. This perpendicular bisector property is due to the symmetrical nature of circles: all radii to the circumference are equal, so the distance from the centre to any point along a chord will be equal.
Example:
In a circle, there's a chord (KL) that doesn't pass through the centre. Draw a line from the centre (O) to this chord (KL) so that it's perpendicular to the chord. If the segment you've created (OKL) is divided into two parts (KM and ML, with M being the point of intersection), then they are equal in length.
3
In a circle where a perpendicular from the centre (O) to a chord (PQ) divides it into two equal parts of 8 cm each, what is the full length of the chord (PQ)?
Circle theorem 4 - Angle in a Semicircle
The Angle in a Semicircle theorem states that any angle inscribed in a semicircle is a right angle.
For example, a circle has a diameter (AB) that divides it into equal halves, creating a semicircle above the diameter. If any point (C) is chosen on the circumference (except the diameter) and lines are drawn from either end of the diameter to this point, then the angle created (ACB) is always a right angle.
Example:
Make a circle with a diameter (XY) and a point (Z) on the circumference that isn't on the diameter. Connecting either end of the diameter and this point creates a triangle (XYZ). Using this theorem, the new angle is 90 degrees, regardless of where the point is on the circumference.
4
If X and Y are endpoints of a diameter and point Z is on the circumference to create a triangle (XYZ), what is this angle (XYZ)?
Circle theorem 5 - Tangent-Secant
This theorem offers critical insights for handling problems involving tangents and their intersections with circle chords. It states that the square of the length of a tangent segment from a point outside a circle to the point of tangency is equal to the product of the lengths of the secant segment and its external segment.
Consider a point A outside a circle from which a tangent and a secant are drawn to the circle. Let the tangent touch the circle at point T and the secant enter the circle at point P and exit at Q. According to the Tangent-Secant Theorem, the square of the length of segment AT (the tangent) is equal to the product of the lengths of the entire secant segment AP and its external part AQ.
Example:
If a tangent AT measures 6 units and a secant line APQ is drawn such that AP measures 9 units and AQ measures 3 units, then the relationship defined by the theorem can be expressed as AT² = AP × AQ. Substituting the given values, we get 6² = 9 × 3, confirming the theorem's correctness.
5
If a tangent from point A to a circle measures 5 units and the external part of a secant from A measures 2 units, with the entire secant measuring 7 units, what is the relationship according to the Tangent-Secant Theorem?
Circle theorem 6 - Cyclic Quadrilateral
A cyclic quadrilateral is where all four vertices are on the circle's circumference. This theorem states that the measurements of 2 opposite angles of a quadrilateral always add up to 180°.
Example:
Let's say we have a quadrilateral (ABCD) inscribed in a circle. If one angle (A) is 70 degrees and another angle (C) is 110 degrees, this fits the theorem as they can be added together to make 180°. Similarly, the opposite angles (B and D) will also sum to 180 degrees.
6
In a cyclic quadrilateral, if one of the opposite angles measures 80 degrees, what is the measure of the other opposite angle?
Circle theorem 7 - Extended Chord
The Extended Chord Theorem is also known as the Secant-Secant theorem. It explores relationships with two secant lines extending from a point outside the circle and intersecting it. The theorem states that the product of the secant line and the external segment for one secant is equal to the product for the other secant.
For example, there are two secants (PA and PB) coming from an external point (P). They intersect a circle at two points (A and B) and continue inside the circle to two points (C and D). The entire length of one secant (including the part outside the circle) is the same as the length of the other secant and its external segment.
7
A person draws two secant lines from point P. The first secant has an external segment of 3 units and a total length of 9 units. The second has an external segment of 2 units and a total length of 8 units. Which calculation is correct?
Circle theorem 8 - Chord-Chord Product
The Chord-Chord theorem states that when two chords intersect in a circle, the products of the lengths of the segments for each chord will be equal.
There are two chords (AB and CD) that intersect at a point in the circle (X). The product of the lengths of the segments of one chord (AB - AX and XB) is equal to the combined lengths of the segments of the other chord (CD - CX and XD).
Example:
The chord AB intersects with chord CD at point X. AX = 3 cm, XB = 2 cm, CX = 4 cm and XD = 5 cm. The product of the lengths of the segments of AB should equal the product of the lengths of the segments of CD: 3 cm × 2 cm = 4 cm × 5 cm.
8
If chord AB intersects chord CD at point X within a circle and the segments are AX = 6 cm, XB = 4 cm, CX = 3 cm and XD = 8 cm, what is the product comparison?
Circle theorem 9 - Equal Chord
The Equal Chord theorem states that chords of equal length are the same distance from the centre. If there are two chords (AB and CD) that are the same equal length, their distance to the centre must be the same. This is because the circle's radius must reach each point of the chords perpendicularly at the midpoint. This ensures each chord is equally positioned in the circle.
Example:
Two chords (AB and CD) are 10 cm long. If the perpendicular distance from the circle's centre (O) to one chord is 3 cm, then the distance to the centre for the other chords is also 3 cm.
9
If two chords, GH and IJ, are 8 cm long and the distance from the centre to chord GH is 2 cm, what is the distance from the centre to chord IJ?
Circle theorem 10 - Radii to Tangents
The Radii to Tangents theorem states that the radii to the points of tangency with tangents drawn from an external point are equal in length. Let's say two tangent lines are drawn from a common external point (P), touching the circle at two points (A and B). The radii that connect the circle's centre (O) to the points of tangency are equal (OA and OB). This equality comes from the properties of tangents: a radius drawn to a point of tangency is perpendicular to the tangent line at that point.
Example:
A point outside a circle (P) has two tangents (PA and PB) touching the circle (A and B) and the radii are drawn to these points (OA and OB). The lengths of these radii must be the same.
10
If two tangents from point R touch a circle at points S and T and the radius to point S is 5 cm, what is the length of the radius to point T?
Circle theorem 11 - Inscribed Angle
The Inscribed Angle theorem details the relationship between an angle inscribed in a circle and the arc it subtends. It states that an angle inscribed in a circle is half the measure of its subtended arc.
Consider a circle with an angle <ABC inscribed by three vertices (A, B, C). The arc subtended by this angle is the part of the circumference between two vertices (A and C), passing through the middle vertice (B). The theorem says that this angle is exactly half the measure of the arc (AC). This property holds for any inscribed angle and can find unknown measures using known arcs.
Example:
If an arc (AC) subtends 120 degrees at the circle's centre, the inscribed angle that subtends the same arc from the circle's edge will measure 60 degrees.
11
If an arc subtends 80 degrees at the circle's centre, what is the inscribed angle measurement that subtends the same arc?
Final thoughts on geometry circle theorems
A circle is not just a shape in geometry but a way to learn about its inner relationships and properties. Understanding circle theorems is used to solve many problems in mathematics, physics, engineering and beyond. For further reading, there is a good discussion with users on Quora about the circle theorems usage in the real world.
To improve your understanding of circle theorems, learning how to solve them using proofs is a good idea. Follow the link to see the steps required and examples of how to prove circle theorems.
We hope this article has been useful. If you need further support revising this topic, TeachTutti has a list of qualified GCSE Maths tutors you can learn with in-person or online.
Glossary
- Circle Theorems - Geometric rules describing the relationships between angles, lines and segments that intersect or form inside/around a circle.
- Tangent - A line that touches a circle, never entering the interior.
- Secant - A line that intersects a circle at two points, passing through the interior.
- Chord - A line segment with endpoints on the circle. It's entirely inside the circle.
- Cyclic Quadrilateral - A quadrilateral where all four vertices are on the circumference.
- Inscribed Angle - An angle formed by two chords in a circle that share an endpoint.
- Central Angle - An angle whose vertex is the circle's centre and whose sides pass through two points on the circle.
- Arc - A portion of the circumference of a circle.
- Radius - A line segment from the circle's centre to any point on its circumference.
- Diameter - A chord that passes through the centre of the circle, the longest possible chord.
This post was updated on 05 Jul, 2024.
