Coordinate Geometry questions involve points, lines, and shapes in the xy coordinate plane. Though common on the ACT, there are only certain formulas that you’ll need to memorize in order to ace the Coordinate Geometry sections.
In fact, some simple Coordinate Geometry questions on the ACT won’t even require you to do any math calculations!
Coordinate Geometry Without a Formula
Take a look at this example:
A triangle, RST, is reflected across the y-axis to form the image triangle R’S’T’ in the standard (x,y) coordinate plane; thus, R reflects R’. The coordinates of point T are (j,k). What are the coordinates of point T’?
(A) (-j, k)
(B) (j, -k)
(C) (-j, -k)
(D) (k, j)
(E) It cannot be determined.
When you reflect a point across the y-axis, the value of its x-value is made negative. So: (j,k) –> (-j,k)
To visualize, draw a picture:
The correct answer is (A).
Basic Coordinate Geometry Formulas
However, the more challenging Math questions on the ACT involve Coordinate Geometry formulas, so it’s important you have a solid grasp of concepts tested. Here are some of the fundamental formulas to know:
Midpoint Formula =
Use this to find the midpoint between two points (notice how you are essentially finding the average of the x-coordinates and the average of the y-coordinates).
Slope = Rise / Run = Change in y / Change in x
As long as you know any two points on a line, you can find the slope. Remember that parallel lines have the same slope, and perpendicular lines have negative reciprocal slopes. You’ll also need to know how to recognize the graphs for linear and non-linear equations.
y = mx + b
This is called slope-intercept form. An equation in this form will always make a straight line on a graph (notice how neither x nor y have an exponent). In this form, b is the y-intercept (the point on the y-axis where the line crosses) and m is the slope.
y = ax 2 + bx + c
This is the standard equation for a parabola. In this equation c represents the y-intercept. A standard equation in which a variable is squared will never make a straight line.
The standard equation of a circle is (x – h)2 + (y – k)2 = r2 where (h, k) is the center point of the circle and r is the radius.