GeometryGeometryHigh school

Coordinate Geometry

Distance, midpoint and slope on the coordinate plane.

Coordinate geometry places shapes on the $xy$ -plane so that geometric questions become algebra. The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ , which is the Pythagorean theorem applied to the horizontal and vertical gaps.

The midpoint averages the coordinates: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ . With distance, midpoint and slope you can prove that a quadrilateral is a parallelogram, find the centre of a circle from a diameter, or show that two segments are perpendicular.

Key terms

Distance formula: The Pythagorean theorem applied to two points.
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint: The average of the coordinates.
$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Key ideas

Distance formula for lengths, midpoint formula for centres, slope for parallel and perpendicular checks.
A shape is a right triangle if two sides have slopes with product $-1$ .
Perpendicular bisector: passes through the midpoint with the negative reciprocal slope.
Sketch the points first to catch sign errors.

Worked example

Show that $A(1, 1)$ , $B(4, 5)$ , $C(8, 2)$ form a right triangle.

Slope $AB = \frac{5 - 1}{4 - 1} = \frac{4}{3}$ .
Slope $BC = \frac{2 - 5}{8 - 4} = -\frac{3}{4}$ .
Product: $\frac{4}{3} \times \left(-\frac{3}{4}\right) = -1$ , so $AB \perp BC$ .
Also $AB = 5$ , $BC = 5$ , $AC = \sqrt{50}$ , and $25 + 25 = 50$ .

Answer: Right angle at $B$

Common mistakes

Subtracting coordinates in different orders inside the distance formula (harmless when squared, but fatal for slope).
Averaging $x$ with $y$ in the midpoint formula.
Forgetting the square root in the distance formula.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Find the distance between $(-2, 3)$ and $(4, 11)$ .
Show answer
$10$
2.Find the $x$ -coordinate of the midpoint of $(-3, 7)$ and $(9, 1)$ .
Show answer
$3$
3.A diameter runs from $(0, 0)$ to $(6, 8)$ . What is the radius?
Show answer
$5$

Coordinate Geometry

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