# IGCSE Mathematics Higher: Circle Theorems

## Scheme of work: IGCSE Higher: Year 10: Term 4: Circle Theorems

#### Prerequisite Knowledge

1. Calculate Interior and Exterior Angles in a Polygon
• Concept: Understanding how to calculate the sum of interior angles in a polygon and finding individual interior and exterior angles.
• Example: $\text{Sum of interior angles of an n-sided polygon: } (n-2) \times 180^\circ.$ $\text{For a hexagon (6 sides): } (6-2) \times 180^\circ = 720^\circ.$ $\text{Each interior angle of a regular hexagon: } \frac{720^\circ}{6} = 120^\circ.$ $\text{Each exterior angle of a regular hexagon: } 180^\circ – 120^\circ = 60^\circ.$
2. Angles in Parallel Lines
• Concept: Understanding corresponding angles, alternate angles, and co-interior angles formed by parallel lines and a transversal.
• Example: $\text{If } \angle A = 50^\circ \text{, then } \angle B = 50^\circ \text{ (corresponding angles)}.$ $\text{If } \angle C = 70^\circ \text{, then } \angle D = 70^\circ \text{ (alternate angles)}.$ $\text{If } \angle E = 110^\circ \text{, then } \angle F = 70^\circ \text{ (co-interior angles, sum to } 180^\circ \text{)}.$
3. Lengths of Similar Shapes
• Concept: Understanding the properties of similar shapes, including the ratio of corresponding side lengths.
• Example: $\text{If two shapes are similar and the ratio of corresponding side lengths is } 2:3 \text{, then all corresponding lengths are in this ratio}.$ $\text{If a side of one shape is } 6 \text{ cm, the corresponding side of the similar shape is } 9 \text{ cm}.$

#### Success Criteria

1. Understand and Use the Internal and External Intersecting Chord Properties
• Objective: Apply the intersecting chord theorem to solve problems involving internal and external intersecting chords.
• Example: $\text{For intersecting chords } AB \text{ and } CD \text{ inside a circle, if } A \text{ and } C \text{ are endpoints of one chord and } B \text{ and } D \text{ are endpoints of the other chord, then } AC \times CB = AD \times DB.$
2. Recognise the Term ‘Cyclic Quadrilateral’
• Objective: Identify and describe the properties of cyclic quadrilaterals.
• Example: $\text{A cyclic quadrilateral is a four-sided figure where all vertices lie on the circumference of a circle.}$
3. Understand and Use Angle Properties of the Circle
• Objective: Apply the angle properties of the circle to solve problems involving circles.
• Example:
• Angle Subtended by an Arc at the Centre is Twice the Angle Subtended at the Circumference: $\text{If } \angle AOB \text{ is the angle at the centre, and } \angle APB \text{ is the angle at the circumference, then } \angle AOB = 2 \times \angle APB.$
• Angle Subtended at the Circumference by a Diameter is a Right Angle: $\text{If } AB \text{ is the diameter, then } \angle ACB = 90^\circ.$
• Angles in the Same Segment are Equal: $\text{If } \angle APB \text{ and } \angle AQB \text{ are in the same segment, then } \angle APB = \angle AQB.$
• Sum of the Opposite Angles of a Cyclic Quadrilateral is 180 Degrees: $\text{If } ABCD \text{ is a cyclic quadrilateral, then } \angle A + \angle C = 180^\circ \text{ and } \angle B + \angle D = 180^\circ.$
• The Alternate Segment Theorem: $\text{If a tangent and a chord meet at a point on a circle, the angle between the tangent and the chord is equal to the angle in the alternate segment.}$

#### Key Concepts

1. Understand and Use the Internal and External Intersecting Chord Properties
• Concept: The intersecting chord theorem states that the products of the segments of two intersecting chords are equal.
• Example: $\text{For intersecting chords } AB \text{ and } CD \text{ inside a circle, } AC \times CB = AD \times DB.$
2. Recognise the Term ‘Cyclic Quadrilateral’
• Concept: A cyclic quadrilateral is a four-sided figure where all vertices lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral sum to 180 degrees.
• Example: $\text{For cyclic quadrilateral } ABCD, \text{ } \angle A + \angle C = 180^\circ \text{ and } \angle B + \angle D = 180^\circ.$
3. Understand and Use Angle Properties of the Circle
• Concept: Several important theorems describe the relationships between angles subtended by arcs, chords, and tangents in a circle.
• Example:
• Angle Subtended by an Arc at the Centre is Twice the Angle Subtended at the Circumference: $\text{If } \angle AOB \text{ is the angle at the centre, and } \angle APB \text{ is the angle at the circumference, then } \angle AOB = 2 \times \angle APB.$
• Angle Subtended at the Circumference by a Diameter is a Right Angle: $\text{If } AB \text{ is the diameter, then } \angle ACB = 90^\circ.$
• Angles in the Same Segment are Equal: $\text{If } \angle APB \text{ and } \angle AQB \text{ are in the same segment, then } \angle APB = \angle AQB.$
• Sum of the Opposite Angles of a Cyclic Quadrilateral is 180 Degrees: $\text{If } ABCD \text{ is a cyclic quadrilateral, then } \angle A + \angle C = 180^\circ \text{ and } \angle B + \angle D = 180^\circ.$
• The Alternate Segment Theorem: $\text{If a tangent and a chord meet at a point on a circle, the angle between the tangent and the chord is equal to the angle in the alternate segment.}$

#### Common Misconceptions

1. Understand and Use the Internal and External Intersecting Chord Properties
• Common Mistake: Misapplying the intersecting chord theorem by confusing which segments to multiply together.
• Example: $\text{Incorrect: For intersecting chords } AB \text{ and } CD, \text{ calculating } AC \times CD \text{ instead of } AC \times CB = AD \times DB.$
2. Recognise the Term ‘Cyclic Quadrilateral’
• Common Mistake: Misidentifying a quadrilateral as cyclic when not all its vertices lie on the circumference of a circle.
• Example: $\text{Incorrect: Assuming any quadrilateral is cyclic without checking if all vertices lie on the circle.}$ $\text{Correct: Verify that all vertices of the quadrilateral lie on the circumference of the circle.}$
3. Understand and Use Angle Properties of the Circle
• Angle Subtended by an Arc at the Centre is Twice the Angle Subtended at the Circumference:
• Common Mistake: Confusing which angle is at the centre and which is at the circumference.
• Example: $\text{Incorrect: Assuming } \angle AOB \text{ is at the circumference and } \angle APB \text{ is at the centre.}$ $\text{Correct: } \angle AOB \text{ is the angle at the centre and } \angle APB \text{ is the angle at the circumference.}$
• Angle Subtended at the Circumference by a Diameter is a Right Angle:
• Common Mistake: Forgetting that the angle subtended by the diameter at the circumference is always a right angle.
• Example: $\text{Incorrect: Assuming } \angle ACB \text{ is not } 90^\circ \text{ when } AB \text{ is the diameter.}$ $\text{Correct: Remember that } \angle ACB = 90^\circ \text{ when } AB \text{ is the diameter.}$
• Angles in the Same Segment are Equal:
• Common Mistake: Not recognizing which angles are in the same segment.
• Example: $\text{Incorrect: Assuming } \angle APB \text{ and } \angle AQD \text{ are equal when they are not in the same segment.}$ $\text{Correct: Ensure that } \angle APB \text{ and } \angle AQB \text{ are in the same segment.}$
• Sum of the Opposite Angles of a Cyclic Quadrilateral is 180 Degrees:
• Common Mistake: Forgetting that this property only applies to cyclic quadrilaterals.
• Example: $\text{Incorrect: Assuming } \angle A + \angle C = 180^\circ \text{ for any quadrilateral.}$ $\text{Correct: This property holds true only if the quadrilateral is cyclic.}$
• The Alternate Segment Theorem:
• Common Mistake: Misidentifying the alternate segment and incorrectly applying the theorem.
• Example: $\text{Incorrect: Assuming the angle between the tangent and the chord is equal to an angle not in the alternate segment.}$ $\text{Correct: The angle between the tangent and the chord is equal to the angle in the alternate segment.}$

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