The coordinate plane, distance and midpoint

Turning geometry into algebra

René Descartes had a simple, powerful idea: describe every point with a pair of numbers, and every shape with an equation. Geometry problems then become algebra problems.

Two perpendicular number lines, the \(x\)-axis (horizontal) and the \(y\)-axis (vertical), cross at the origin \((0, 0)\). Every point is an ordered pair \((x, y)\): first how far to move horizontally, then vertically.

The axes split the plane into four quadrants, numbered counterclockwise starting from the upper right:

Points on the axes don't belong to any quadrant.

The distance formula

Between \((x_1, y_1)\) and \((x_2, y_2)\), draw a right triangle with horizontal and vertical legs. The legs have lengths \(|x_2 - x_1|\) and \(|y_2 - y_1|\), and the segment you want is the hypotenuse. The Pythagorean theorem gives:

The midpoint formula

Classifying a triangle with coordinates

To check for a right triangle, compare \(c^2\) with \(a^2 + b^2\) using the squared distances, so you can skip the square roots.

Practice