# Circles and Chords in A-Level Mathematics

Understanding the properties of circles and chords is a fundamental part of coordinate geometry in A-Level mathematics. In these tutorials, I demonstrate different methods to find the equations of circles and related lines. Each tutorial builds on the previous one, to show progression.

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Members of Mr Mathematics can download the full lesson and accompanying student handout. These resources provide additional practice problems and detailed solutions to reinforce your understanding of circles and chords in A-Level mathematics.

## Part 1: Finding the Equation of a Line through a Circle’s Chord

Skills Learned:

• Calculating the midpoint of a line segment.
• Determining the equation of the perpendicular bisector of a chord.
• Using coordinate geometry to verify geometric properties, such as a line passing through the centre of a circle.

## Part 2: Perpendicular Bisector of a Chord

Learning Progression:

• Building on the concept of finding the midpoint, this tutorial introduces calculating the gradient of a line segment.
• Develops skills in determining the equation of the perpendicular bisector of a chord using the gradient.
• Enhances understanding of the properties of a circle by proving that the perpendicular bisector passes through the circle’s centre.

## Part 3: Equation of a Circle with a Given Chord and Tangent

Learning Progression:

• Extends knowledge by integrating the concept of a tangent line to a circle.
• Focuses on deriving the circle’s center and radius using geometric relationships between the tangent, chord, and circle.
• Combines previously learned skills to form and solve the equation of a circle with more complex conditions.

## Lesson Plan and Resources

The downloadable lesson includes a PowerPoint presentation with a starter, main, and plenary sections. A differentiated worksheet provides additional examples, ensuring students at all levels have sufficient practise. Solutions are included to facilitate self-assessment and correction.

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