Coordinate Geometry

7 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team

Coordinate geometry places shapes on a grid so you can describe them with numbers. Every point has two coordinates (x, y), and simple formulas give you distances, midpoints and slopes. It is the bridge between geometry and algebra.

The coordinate plane

The horizontal x-axis and vertical y-axis cross at the origin (0, 0). They divide the plane into four quadrants: I (+, +), II (−, +), III (−, −) and IV (+, −). To plot (3, −2), move 3 right and 2 down.

Distance between two points

d = √((x₂ − x₁)² + (y₂ − y₁)²), which is the Pythagorean theorem. For A(−2, 1) and B(4, 9): d = √(6² + 8²) = √100 = 10.

Midpoint

Average the coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2). For A and B above, the midpoint is (1, 5).

Slope

m = (y₂ − y₁) ÷ (x₂ − x₁) = rise over run. For A and B: 8/6 = 4/3. Positive slopes rise to the right, negative slopes fall, and a horizontal line has slope 0. A vertical line has undefined slope.

Equation of a line

Use y − y₁ = m(x − x₁). Through A(−2, 1) with slope 4/3: y − 1 = (4/3)(x + 2), so y = (4/3)x + 11/3. Plot it in the graphing calculator.

Reflections

Reflecting a point over the x-axis changes (x, y) to (x, −y). Over the y-axis, it becomes (−x, y). Through the origin, it becomes (−x, −y).

Practice questions

Test yourself, then tap to check. Answers are calculated by the tools linked below.

Find the roots of f(x) = x² − 6x + 8.
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Function: f(x) = x^2 - 6x + 8
Solve 4x − 3 = x + 12.
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x: 5
A right triangle has legs 9 and 12. Find the hypotenuse.
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Hypotenuse c: 15