Coordinate Geometry Formulas Card Cheat Sheet

A printable reference covering distance, midpoint, slope, line equations, parallel and perpendicular lines, section formula, and circle equations for grades 7-11.

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Coordinate geometry connects algebra and geometry by placing points, lines, and shapes on the coordinate plane. This formulas card helps students quickly find distances, midpoints, slopes, equations of lines, and circle equations. It is useful for graphing, proving geometric relationships, and solving problems involving coordinates. Grades 7-11 students can use it as a quick reference during practice or 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 two points is $m=\frac{y_2-y_1}{x_2-x_1}$ , where $x_2\ne x_1$ .
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.
Standard form of a line is $Ax+By=C$ , where $A$ , $B$ , and $C$ are constants and $A$ and $B$ are not both $0$ .
Parallel nonvertical lines have equal slopes, so $m_1=m_2$ , and perpendicular nonvertical lines have slopes with product $m_1m_2=-1$ .
The circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$ .

Vocabulary

Coordinate Plane: A flat grid formed by the $x$ -axis and $y$ -axis where points are located using ordered pairs.
Ordered Pair: A pair $(x,y)$ that gives the horizontal position $x$ and vertical position $y$ of a point.
Slope: The rate of change of a line, found by $m=\frac{\Delta y}{\Delta x}$ for a nonvertical line.
Intercept: An intercept is a point where a graph crosses an axis, such as the $y$ -intercept $(0,b)$ in $y=mx+b$ .
Midpoint: The midpoint is the point exactly halfway between two endpoints on a segment.
Radius: The radius is the distance from the center of a circle to any point on the circle.

Common Mistakes to Avoid

Reversing the coordinates in the slope formula is wrong because both numerator and denominator must follow the same point order, such as $m=\frac{y_2-y_1}{x_2-x_1}$ .
Forgetting the square root in the distance formula is wrong because $(x_2-x_1)^2+(y_2-y_1)^2$ gives $d^2$ , not $d$ .
Averaging only the $x$ -coordinates for a midpoint is wrong because the midpoint requires both coordinates: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ .
Saying every vertical line has slope $0$ is wrong because vertical lines have undefined slope, while horizontal lines have slope $0$ .
Using the same slope for perpendicular lines is wrong because perpendicular nonvertical lines have negative reciprocal slopes, so $m_1m_2=-1$ .

Practice Questions

1 Find the distance and midpoint between $A(-3,4)$ and $B(5,-2)$ .
2 Find the slope of the line through $(2,-1)$ and $(8,11)$ , then write its equation in point-slope form.
3 Write the equation of a circle with center $(3,-5)$ and radius $4$ .
4 A line has slope $\frac{2}{3}$ . Explain how to identify the slope of a line parallel to it and the slope of a line perpendicular to it.

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