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Coordinate Geometry: Distance, Midpoint & Slope
Every point on the coordinate plane has an address: (x, y). Once two points have addresses, you can measure the line segment between them without a ruler — just arithmetic. Three tools do almost all the work: how far apart the points are, the middle point between them, and how tilted the line is.
The three formulas
For points (x₁, y₁) and (x₂, y₂), think of the horizontal gap (x₂ − x₁) and the vertical gap (y₂ − y₁).
Distance comes straight from the Pythagorean theorem: the two gaps are the legs of a right triangle, and the segment is the hypotenuse. Midpoint just averages the x's and averages the y's. Slope is "rise over run" — vertical change divided by horizontal change.
How to use them
Label your two points so you know which is (x₁, y₁).
Subtract in the same order for both x and y.
Keep signs — a negative slope means the line falls left to right.