Circle Theorems - A comprehensive guide to essential geometry rules with visual examples.
A colorful educational poster displaying six key circle theorems with diagrams and explanations, including the alternate segment theorem, angle at the centre theorem, angles in the same segment theorem, angles in a semicircle, chord of a circle, cyclic quadrilateral, and tangent of a circle.
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Step-by-step solution for: Circle Theorems - GCSE Maths - Steps, Examples & Worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Circle Theorems - GCSE Maths - Steps, Examples & Worksheet
The image you provided summarizes several key theorems related to circles in geometry. These theorems are fundamental for solving problems involving circles, tangents, chords, and angles. Below, I will explain each theorem in detail and provide a brief example of how they can be applied.
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- Statement: The angle that lies between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.
- Diagram: In the diagram, the angle between the tangent at point \( A \) and the chord \( AC \) (angle \( \theta \)) is equal to the angle subtended by the chord \( AC \) in the alternate segment (also \( \theta \)).
- Application Example:
- If the angle between the tangent and the chord is given as \( 30^\circ \), then the angle subtended by the chord in the alternate segment is also \( 30^\circ \).
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- Statement: The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.
- Diagram: In the diagram, the angle at the centre \( \angle AOB = 2x \) is twice the angle at the circumference \( \angle ACB = x \).
- Application Example:
- If the angle at the circumference is \( 40^\circ \), then the angle at the centre is \( 2 \times 40^\circ = 80^\circ \).
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- Statement: Angles in the same segment of a circle are equal.
- Diagram: In the diagram, the angles \( \angle ABC \) and \( \angle ADC \) are both subtended by the same arc \( AC \), so they are equal.
- Application Example:
- If \( \angle ABC = 50^\circ \), then \( \angle ADC = 50^\circ \) because they are in the same segment.
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- Statement: The angle in a semicircle is \( 90^\circ \).
- Diagram: In the diagram, the angle \( \angle ACB \) is a right angle because it is subtended by the diameter \( AB \).
- Application Example:
- If \( AB \) is the diameter of the circle, then any angle subtended by \( AB \) at the circumference (e.g., \( \angle ACB \)) is \( 90^\circ \).
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- Statement: The perpendicular from the centre of a circle to a chord bisects the chord.
- Diagram: In the diagram, the perpendicular from the centre \( O \) to the chord \( AB \) splits it into two equal parts, \( AD = DB \).
- Application Example:
- If the length of the chord \( AB \) is \( 10 \) units and the perpendicular from the centre bisects it, then each half of the chord is \( 5 \) units.
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- Statement: The opposite angles in a cyclic quadrilateral total \( 180^\circ \).
- Diagram: In the diagram, the opposite angles \( \angle A + \angle C = 180^\circ \) and \( \angle B + \angle D = 180^\circ \).
- Application Example:
- If one pair of opposite angles in a cyclic quadrilateral are \( 70^\circ \) and \( 110^\circ \), then the other pair must also sum to \( 180^\circ \).
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- Part A: The angle between a tangent and a radius at the point of contact is \( 90^\circ \).
- Diagram: In Diagram A, the tangent at point \( A \) is perpendicular to the radius \( OA \).
- Application Example:
- If a radius meets a tangent at point \( A \), the angle formed is always \( 90^\circ \).
- Part B: Tangents drawn from an external point to a circle are equal in length.
- Diagram: In Diagram B, the tangents \( AD \) and \( AB \) from point \( A \) to the circle are equal in length.
- Application Example:
- If \( AD = 6 \) units, then \( AB \) is also \( 6 \) units.
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1. Alternate Segment Theorem: Angles between a tangent and a chord are equal to the angle in the alternate segment.
2. Angle at the Centre Theorem: The angle at the centre is twice the angle at the circumference.
3. Angles in the Same Segment Theorem: Angles in the same segment are equal.
4. Angles in a Semicircle: The angle in a semicircle is \( 90^\circ \).
5. Chord of a Circle: The perpendicular from the centre bisects the chord.
6. Cyclic Quadrilateral: Opposite angles sum to \( 180^\circ \).
7. Tangent of a Circle:
- The angle between a tangent and radius is \( 90^\circ \).
- Tangents from an external point are equal in length.
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\[
\boxed{\text{See detailed explanations above for each theorem.}}
\]
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1. Alternate Segment Theorem
- Statement: The angle that lies between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.
- Diagram: In the diagram, the angle between the tangent at point \( A \) and the chord \( AC \) (angle \( \theta \)) is equal to the angle subtended by the chord \( AC \) in the alternate segment (also \( \theta \)).
- Application Example:
- If the angle between the tangent and the chord is given as \( 30^\circ \), then the angle subtended by the chord in the alternate segment is also \( 30^\circ \).
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2. Angle at the Centre Theorem
- Statement: The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.
- Diagram: In the diagram, the angle at the centre \( \angle AOB = 2x \) is twice the angle at the circumference \( \angle ACB = x \).
- Application Example:
- If the angle at the circumference is \( 40^\circ \), then the angle at the centre is \( 2 \times 40^\circ = 80^\circ \).
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3. Angles in the Same Segment Theorem
- Statement: Angles in the same segment of a circle are equal.
- Diagram: In the diagram, the angles \( \angle ABC \) and \( \angle ADC \) are both subtended by the same arc \( AC \), so they are equal.
- Application Example:
- If \( \angle ABC = 50^\circ \), then \( \angle ADC = 50^\circ \) because they are in the same segment.
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4. Angles in a Semicircle
- Statement: The angle in a semicircle is \( 90^\circ \).
- Diagram: In the diagram, the angle \( \angle ACB \) is a right angle because it is subtended by the diameter \( AB \).
- Application Example:
- If \( AB \) is the diameter of the circle, then any angle subtended by \( AB \) at the circumference (e.g., \( \angle ACB \)) is \( 90^\circ \).
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5. Chord of a Circle
- Statement: The perpendicular from the centre of a circle to a chord bisects the chord.
- Diagram: In the diagram, the perpendicular from the centre \( O \) to the chord \( AB \) splits it into two equal parts, \( AD = DB \).
- Application Example:
- If the length of the chord \( AB \) is \( 10 \) units and the perpendicular from the centre bisects it, then each half of the chord is \( 5 \) units.
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6. Cyclic Quadrilateral
- Statement: The opposite angles in a cyclic quadrilateral total \( 180^\circ \).
- Diagram: In the diagram, the opposite angles \( \angle A + \angle C = 180^\circ \) and \( \angle B + \angle D = 180^\circ \).
- Application Example:
- If one pair of opposite angles in a cyclic quadrilateral are \( 70^\circ \) and \( 110^\circ \), then the other pair must also sum to \( 180^\circ \).
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7. Tangent of a Circle
- Part A: The angle between a tangent and a radius at the point of contact is \( 90^\circ \).
- Diagram: In Diagram A, the tangent at point \( A \) is perpendicular to the radius \( OA \).
- Application Example:
- If a radius meets a tangent at point \( A \), the angle formed is always \( 90^\circ \).
- Part B: Tangents drawn from an external point to a circle are equal in length.
- Diagram: In Diagram B, the tangents \( AD \) and \( AB \) from point \( A \) to the circle are equal in length.
- Application Example:
- If \( AD = 6 \) units, then \( AB \) is also \( 6 \) units.
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Summary of Theorems
1. Alternate Segment Theorem: Angles between a tangent and a chord are equal to the angle in the alternate segment.
2. Angle at the Centre Theorem: The angle at the centre is twice the angle at the circumference.
3. Angles in the Same Segment Theorem: Angles in the same segment are equal.
4. Angles in a Semicircle: The angle in a semicircle is \( 90^\circ \).
5. Chord of a Circle: The perpendicular from the centre bisects the chord.
6. Cyclic Quadrilateral: Opposite angles sum to \( 180^\circ \).
7. Tangent of a Circle:
- The angle between a tangent and radius is \( 90^\circ \).
- Tangents from an external point are equal in length.
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Final Answer
\[
\boxed{\text{See detailed explanations above for each theorem.}}
\]
Parent Tip: Review the logic above to help your child master the concept of degrees of a circle worksheet.
