Coordinate Geometry summary

Key learning points

• Distance $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$; midpoint averages coordinates.
• Gradient $m=\frac{y_2-y_1}{x_2-x_1}$; perpendicular lines satisfy $m_1 m_2=-1$.
• Point-gradient form: $y-y_0=m(x-x_0)$.
• Vertical lines $x=k$ have undefined gradient.

Worked example 1 — Perpendicular gradient

A line has gradient $\tfrac{2}{3}$. Find a perpendicular gradient.

Working

Negative reciprocal: flip and change sign.

Worked example 2 — Distance ; midpoint averages coordinates

Use this method from the Coordinate Geometry sheet: Distance $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$; midpoint averages coordinates.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — Gradient ; perpendicular lines satisfy

Use this method from the Coordinate Geometry sheet: Gradient $m=\frac{y_2-y_1}{x_2-x_1}$; perpendicular lines satisfy $m_1 m_2=-1$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Point-gradient form

Use this method from the Coordinate Geometry sheet: Point-gradient form: $y-y_0=m(x-x_0)$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Vertical lines have undefined gradient

Use this method from the Coordinate Geometry sheet: Vertical lines $x=k$ have undefined gradient.

Working

Each key learning point matches a question type on the printable worksheet.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.