History

Coordinate geometry was formalized by Rene Descartes and Pierre de Fermat. Their work connected algebraic equations to geometric curves, creating a major foundation for modern mathematics.

Derivation and reasoning

Why the distance formula works

Horizontal change is \(\Delta x=x_2-x_1\), vertical change is \(\Delta y=y_2-y_1\). These form a right triangle, so distance is \(\sqrt{(\Delta x)^2+(\Delta y)^2}\).

Checkpoints

State the quadrant of \((-4,-7)\).

Answer: Quadrant III.

Find distance between \((1,2)\) and \((7,10)\).

Answer: \(\sqrt{6^2+8^2}=\sqrt{100}=10\).

Find midpoint of \((-3,5)\) and \((9,-1)\).

Answer: \(\left(3,2\right)\).

Find slope through \((2,-1)\) and \((6,7)\).

Answer: \(m=\frac{7-(-1)}{6-2}=\frac84=2\).

Line with slope \(3\) through \((1,4)\): find equation.

Answer: \(y-4=3(x-1)\Rightarrow y=3x+1\).

Applications

• Maps/GPS: locations represented using coordinate systems.
• Computer graphics: points and lines placed on pixel coordinate planes.
• Physics: trajectories and motion graphs depend on coordinates and slope.

References

• Standard high-school coordinate geometry units.
• Analytic geometry references on points, lines, distance, and midpoint.