Coordinate Geometry

Coordinate geometry uses algebraic formulas for slope, distance, midpoint, and lines to analyze shapes on the plane.

A segment’s horizontal and vertical changes form a right triangle for distance and midpoint reasoning. Two coordinate points are joined by a segment with dashed rise and run legs. The midpoint is marked halfway along the segment, connecting coordinate averages with the Pythagorean distance calculation.

Coordinates turn geometric relationships into quantities we can calculate and verify.

The coordinate plane

An ordered pair $(x,y)$ locates a point: move horizontally by $x$, then vertically by $y$. The axes divide the plane into four quadrants, with signs $(+,+)$, $(-,+)$, $(-,-)$, and $(+,-)$ counterclockwise from Quadrant I.

Order matters: $(2,-5)$ and $(-5,2)$ are different points.

Distance formula

The Pythagorean theorem gives

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.

Distance is nonnegative and independent of endpoint order because both changes are squared.

Midpoint formula

The midpoint averages coordinates:

M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).

It lies halfway along the segment in both coordinate directions.

Worked example

One pair of points supplies direction, length, and centre information.

Transformations

Translations add fixed values to coordinates; reflections change signs or swap coordinates depending on the mirror line; rotations follow coordinate rules around the origin.

Coordinate transformations preserve or change length, angle, and orientation in predictable ways.

Common mistakes

Swapping $x$ and $y$. Ordered pairs always list horizontal first.

Reversing subtraction in only one slope component. Keep endpoint order consistent.

Using negative distance. Distance is a magnitude.

Calling equal slopes perpendicular. Equal slopes are parallel.

Classifying a shape from appearance. Use calculated evidence.