A-level maths / Pure 1 / Circles
Circles
Use distances and perpendicular lines to connect circle equations with geometry.
Choose the method you need. Each lesson keeps the written explanation in view and lets you operate the model and optional animation yourself.
- Midpoints and perpendicular bisectorsFind a midpoint, recover a missing endpoint and construct perpendicular bisectors. Understand why every point on a bisector is equally far from the two endpoints.
- Equations of circlesFind a circle equation from its centre, radius or diameter. Complete the square to recover the centre and radius, and test whether a point is inside, on or outside a circle.
- Circle and line intersectionsSolve circle and line equations simultaneously, use the discriminant to count intersections and find tangent conditions. Includes chord lengths and an extension on two intersecting circles.
- Tangents to circlesFind the equation of a tangent to a circle at a point, tangents with a given gradient and tangents from an external point. Use perpendicular radii, discriminants and exact geometry.
- Circles and trianglesUse the angle in a semicircle, recognise a diameter and find a circle through three points. Construct a circumcentre with perpendicular bisectors and verify geometric proofs.
Useful foundations
Review straight-line graphs for gradients and distances, completing the square for circle equations, or the discriminant for intersection counts.