Coordinate Geometry Cheat Sheet

A printable reference covering distance, midpoint, slope, line equations, parallel and perpendicular lines, and coordinate proofs for grades 9-10.

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Coordinate geometry connects algebra and geometry by using ordered pairs, graphs, and equations to describe shapes and relationships. This cheat sheet helps students quickly find the formulas needed to measure segments, analyze lines, and prove geometric facts on the coordinate plane. It is useful for solving problems involving slope, distance, midpoint, and equations of lines. Students in grades 9-10 use these tools often in geometry proofs and algebra review.

Key Facts

The distance between $A(x_1, y_1)$ and $B(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
The midpoint of $A(x_1, y_1)$ and $B(x_2, y_2)$ is $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ .
The slope of a nonvertical line through $A(x_1, y_1)$ and $B(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the $y$ -intercept.
Point-slope form is $y - y_1 = m(x - x_1)$ , which is useful when you know one point and the slope.
Parallel nonvertical lines have equal slopes, so $m_1 = m_2$ .
Perpendicular nonvertical lines have slopes whose product is $-1$ , so $m_1m_2 = -1$ .
A coordinate proof uses formulas such as slope, distance, and midpoint to prove geometric relationships from coordinates.

Vocabulary

Coordinate Plane: A flat grid formed by the $x$ -axis and $y$ -axis where points are located by ordered pairs.
Ordered Pair: A pair of numbers $(x, y)$ that gives the horizontal and vertical location of a point.
Slope: The steepness of a line, found by the ratio $m = \frac{\text{change in } y}{\text{change in } x}$ .
Midpoint: The point that divides a segment into two congruent parts.
Distance Formula: A formula used to find the length of a segment between two coordinate points.
Coordinate Proof: A proof that uses coordinate formulas and algebra to justify a geometric conclusion.

Common Mistakes to Avoid

Subtracting coordinates in different orders, such as using $y_2 - y_1$ but $x_1 - x_2$ , is wrong because slope requires the same point order in numerator and denominator.
Forgetting the square root in the distance formula is wrong because $(x_2 - x_1)^2 + (y_2 - y_1)^2$ gives the square of the distance, not the distance.
Confusing midpoint with distance is wrong because midpoint averages coordinates, while distance uses squared differences and a square root.
Using the same slope for perpendicular lines is wrong because perpendicular nonvertical lines need opposite reciprocal slopes, so $m_1m_2 = -1$ .
Treating a vertical line as having slope $0$ is wrong because a vertical line has an undefined slope, while a horizontal line has slope $0$ .

Practice Questions

1 Find the distance between $A(2, -3)$ and $B(8, 5)$ .
2 Find the midpoint of the segment with endpoints $C(-4, 7)$ and $D(6, -1)$ .
3 Write the equation of the line through $(3, 2)$ with slope $m = -\frac{4}{3}$ in point-slope form.
4 Explain how slope and distance could be used to prove that a quadrilateral on the coordinate plane is a rectangle.