Coordinate Geometry
The Coordinate Geometry (Cartesian Coordinate Plane) questions test coordinate geometry concepts such as Number Line Graphs; Equation of a Line; Slope; Parallel and Perpendicular Lines; and Distance and Midpoint Formulas.
1. Number Line Graphs
The most basic type of graphing is graphing on a number line. For the most part, you will be asked to graph inequalities. Below are four of the most common types of problems you will be asked to graph on the ACT Mathematics Test:
If the inequality sign specifies “greater than or equal to” (≥), or “less than or equal to” (≤), you would use a closed circle instead of an open circle on the designated number or the number line.
2. Equation of a Line
There are three forms used to write an equation of a line. The standard form of an equation of a line is in the form Ax + By = C. This can be transformed into the slope-intercept form of y = mx + b, where m is the slope of the line and b is the y-intercept (that is, the point at which the graph of the line crosses the y-axis). The third form is point-slope form, which is (y − y1) = m(x − x1), where m is the slope and (x1, y1) is a given point on the line.
The ACT Mathematics Test will often require you to put the equation of a line into the slope-intercept form to determine either the slope or the y-intercept of a line as follows:
What is the slope of the line given by the equation 3x + 7y − 16 = 0?
Follow these steps to solve:
3x + 7y − 16 = 0
Isolate y on the left side of the equation.
7y = −3x + 16
y = −3x/7 + 16/7
The slope of the line is −3/7.
3. Slope
The slope of a line is the grade at which the line increases or decreases. Commonly defined as “rise over run,” the slope is a value that is calculated by taking the change in y-coordinates divided by the change in x-coordinates for any two given points on a line. The formula for slope is:
where (x1, y1) and (x2, y2) are the two given points.
For example, if you are given (3, 2) and (5, 6) as two points on a line, the slope would be m = (6 - 2) / (5 - 3) = 4/2 = 2.
A positive slope means the graph of the line will go up and to the right. A negative slope means the graph of the line will go down and to the right. A horizontal line has slope 0, and a vertical line has undefined slope.
4. Parallel and Perpendicular Lines
Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the slope of either of the lines is the negative reciprocal of the slope of the other line.
For example, if the slope of line a is 5, then the slope of line b must be -1/5 for lines a and b to be perpendicular.
5. Distance and Midpoint Formulas
To find the distance between two points on a coordinate graph, use the formula:
where (x1, y1) and (x2, y2) are the two given points.
For example, the distance between (3, 2) and (7, 6) is √[(7 - 3)2 + (6 - 2)2] = √[(4)2 + (4)2] = 4√2
To find the midpoint of a line given two points on the line, use the formula:
For example, the midpoint between (5, 4) and (9, 2) is (5+9)/2 , (4+2)/2 = 7, 3