Circle Geometry

Circle Geometry Basics

• Recognising a circle as a geometric figure where all points are an equal distance from a fixed centre point.
• Learning the key terms related to circles such as the radius (distance from the centre of the circle to any point on the circle), diameter (twice the radius), circumference (the perimeter of the circle), and the arc (part of the outer edge of the circle).
• Comprehending the definition of a sector (portion of a circle defined by two radii and the arc between them) and a segment (area defined by a chord and the arc between the chord’s endpoints).

Equation of a Circle

• Understanding the standard form of the equation of a circle (x-a)² + (y-b)² = r², where (a, b) is the centre of the circle and r is the radius of the circle.
• Learning to derive the equation of a circle from given information such as the radius and centre coordinates.
• Identifying the centre and the radius of a circle from its equation.

Tangents and Normal Lines

• Identifying the tangent to a circle as a line that touches the circle at exactly one point. This point is known as the point of contact.
• Recognising that a tangent to a circle is perpendicular to the radius of the circle at the point of contact.
• Understanding the concept of normal lines which are lines that pass through the point of contact and are perpendicular to the tangent.

Angles in Circles

• Familiarising with the properties of angles formed in a circle, such as angles in the same segment (angles subtended by the same arc or chord are equal), angles at the centre (the angle at the centre of a circle is twice the angle at the circumference, when subtended by the same arc or chord), and the alternate segment theorem (the angle between a tangent and a chord is equal to the angle in the alternate segment).
• Utilising these properties to solve problems involving angles in circles.

Lengths in Circles

• Applying Pythagoras’ theorem and the cosine rule to find unknown lengths in circle geometry problems.
• Calculating the length of an arc using the formula Arc Length = rθ, where r is the radius and θ is the angle between the radii of the sector.
• Finding the area of a sector using the formula Sector Area = 0.5r²θ, where r is the radius and θ is the angle between the radii of the sector.

Chords and Tangent Properties

• Comprehending the properties of perpendicular bisectors of chords, namely, that they pass through the centre of the circle.
• Recognising the alternate segment rule where the angle between a tangent and a chord is equal to the angle in the opposite segment.
• Acquiring understanding of the tangent properties such as: tangents from a common point are equal in length, and that the triangle formed by the centre of the circle, the point outside the circle and the point of tangency is right angled.