Coordinate geometry is geometry dealing primarily with
the line graphs and the (x, y)
coordinate plane. The ACT Math Test includes nine questions on coordinate
geometry. The topics you need to know are:
Number Lines and Inequalities
- The (x,y) Coordinate
and Perpendicular Lines
Equation of a Line
Most of the questions on coordinate geometry focus on
slope. About two questions on each test will cover number lines
and inequalities. The other topics are usually covered by just one
question, if they are covered at all.
Number Lines and Inequalities
Number line questions generally ask you to graph inequalities.
A typical number line graph question will ask you:
is the graph of the solution set for 2(x + 5) >
To answer this question, you first must solve for x.
Divide both sides by 2 to get: x +
5 > 2
- Subtract 5 from
both sides to get: x > –3
you match x > –3 to its line graph:
The circles at the endpoints of a line indicating an inequality
are very important when trying to match an inequality to a line
graph. An open circle at –3 denotes that x is
greater than but not equal to –3.
A closed circle would have indicated that x is
greater than or equal to –3.
For the solution set –3 < x <
3, where x must be greater
than –3 and less than 3, the line graph
looks like this:
The (x,y) Coordinate Plane
The (x,y) coordinate plane
is described by two perpendicular lines, the x-axis
and the y-axis. The intersection of
these axes is called the origin. The location of any other point
on the plane (which extends in all directions without limit) can
be described by a pair of coordinates. Here is a figure of the coordinate
plane with a few points drawn in and labeled with their coordinates:
As you can see from the figure, each of the
points on the coordinate plane receives a pair of coordinates: (x,y). The
first coordinate in a coordinate pair is called the x-coordinate.
The x-coordinate of a point
is its location along the x-axis and
can be determined by the point’s distance from the y-axis
(x = 0 at the y-axis).
If the point is to the right of the y-axis,
its x-coordinate is positive,
and if the point is to the left of the y-axis,
its x-coordinate is negative.
The second coordinate in a coordinate pair is the y-coordinate.
The y-coordinate of a point is its
location along the y-axis and can
be calculated as the distance from that point to the x-axis.
If the point is above the x-axis,
its y-coordinate is positive; if the
point is below the x-axis, its y-coordinate
The ACT often tests your understanding of the coordinate
plane and coordinates by telling you the coordinates of the vertices
of a defined geometric shape like a square, and asking you to pick
the coordinates of the last vertex. For example:
the standard (x,y) coordinate
plane, 3 corners of a square are (2,–2), (–2,–2), and (–2,2). What
are the coordinates of the square’s fourth corner?
The best way to solve this sort of problem is to draw
a quick sketch of the coordinate plane and the coordinates given.
You’ll then be able to see the shape described and pick out the coordinates
of the final vertex from the image. In this case, the sketch would
look like this:
A square is the easiest geometric shape which a question
might concern. It is possible that you will be asked to deal with
rectangles or right triangles. The method for any geometric shape
is the same, though. Sketch it out so you can see it.
The ACT occasionally asks test takers to measure the distance
between two points on the coordinate plane. Luckily, measuring distance
in the coordinate plane is made easy thanks to the Pythagorean theorem.
If you are given two points,
distance will always be given by the following formula:
The distance between two points can be represented by
the hypotenuse of a right triangle whose legs are of lengths
following diagram shows how the formula is based on the Pythagorean
theorem (see p. ).
Here’s a sample problem:
the distance between (4,–3) and (–3,8).
To solve this problem, just plug the proper numbers into
the distance formula:
The distance between the points is
equals approximately 13.04
Like finding the distance between two points, the midpoint
between two points in the coordinate plane can be calculated using
a formula. If the endpoints of a line segment are
the midpoint of the line segment is:
In other words, the x- and y-coordinates
of the midpoint are the averages of the x-
and y-coordinates of the endpoints.
Here is a practice question:
is the midpoint of the line segment whose endpoints are (6,0) and
The slope of a line is a measurement of how steeply the
line climbs or falls as it moves from left to right. More technically,
the slope is a line’s vertical change divided by its horizontal change,
also known as “rise over run.” Given two points on a line,
slope of that line can be calculated using the following formula:
The variable most often used to represent slope is m.
So, for example, the slope of a line that contains the
points (–2,–4) and (6,1) is:
Positive and Negative Slopes
You can easily determine whether the slope of a line is
positive or negative just by looking at the line. If a line slopes
uphill as you trace it from left to right, the slope is positive.
If a line slopes downhill as you trace it from left to right, the
slope is negative.
You can determine the relative magnitude of the slope
by the steepness of the line. The steeper the line, the more the
“rise” will exceed the “run,” and the larger
consequently, the slope will be. Conversely, the flatter the line,
the smaller the slope will be.
For practice, look at the lines in the figure below and
try to determine whether their slopes are positive or negative and
which have greater relative slopes:
Lines l and m have
positive slopes, and lines n and o have
negative slope. In terms of slope magnitude, line l
> m > n > o.
It can be helpful to recognize a few slopes by sight.
- A line that is horizontal has a slope of 0.
Since there is no “rise,” and thus
- A line that is vertical has an undefined slope. In this
case, there is no “run,” and Thus and any fraction with 0 in
its denominator is, by definition, undefined.
- A line that makes a angle with a horizontal
has a slope of 1 or –1. This makes sense
because the “rise” equals the “run,” and or
has slope 0
it is horizontal. Line b
has slope –1
it makes a
angle with the
horizontal and slopes downward as you move from left to right. Line c
because it makes a
with the horizontal and slopes upward as you move from left to right.
has undefined slope because it is vertical.
Parallel and Perpendicular Lines
Parallel lines are lines that don’t intersect. In other
words, parallel lines are lines that share the exact same slope.
Perpendicular lines are lines that intersect at a right
angle (or 90%). In coordinate geometry, perpendicular
lines have negative reciprocal slopes. That is, a line with slope m is perpendicular
to a line with a slope of –1/m.
In the figure below are three lines. Lines q and r both
have a slope of 2, so they are parallel. Line s is
perpendicular to both lines q and r,
and thus has a slope of –1/2.
On the ACT, never assume that two lines are parallel or
perpendicular just because they look that way in a diagram. If the
lines are parallel or perpendicular, the ACT will tell you so. (Perpendicular
lines can be indicated by a little square located at the place of
intersection, as in the diagram above.)
Equation of a Line
We’ve already shown you how to find the slope of a line
using two points on the line. It is also possible to find the slope
of a line using the equation of the line. In addition, the equation
of a line can help you find the x-
and y-intercepts of the line, which
are the locations where the line intersects with the x-
and y-axes. This equation for a line
is called the slope-intercept form:
where m is the slope of
the line, and b is the y-intercept
of the line.
Finding the Slope Using the Slope-Intercept Form
If you are given the equation of a line that matches the
slope-intercept form, you immediately know that the slope is equal
to the value of m. However, it is
more likely that the ACT will give you an equation for a line that
doesn’t exactly match the slope-intercept form and ask you to calculate
the slope. In this case, you will have to manipulate the given equation
until it resembles the slope-intercept form. For example,
is the slope of the line defined by the equation 5x +
3y = 6?
To answer this question, isolate the y so
that the equation fits the slope-intercept form.
The slope of the line is –5/3.
Finding the Intercepts Using the Slope-Intercept
The y-intercept of a line
is the y-coordinate of the point at
which the line intersects the y-axis.
Likewise, the x-intercept of a line
is the x-coordinate of the point at
which the line intersects the x-axis.
In order to find the y-intercept,
simply set x = 0 and solve for the
value of y. To find the x-intercept,
set y = 0 and solve for x.
To sketch a line given in slope-intercept form, first
plot the y-intercept, and then use
the slope of the line to plot another point. Connect the two points
to form your line. In the figure below, the line y =
–2x + 3 is graphed.
Since the slope is equal to –2, the line
descends two units for every one unit it moves in the positive x direction.
The y-intercept is at 3,
so the line crosses the y-axis at (0,3).
For the ACT Math Test, you should know how the graphs
of certain equations look. The two equations that are most important
in terms of graphing are
If you add lesser-degree terms to the equations, these
graphs will shift around the origin but retain their basic shape.
You should also keep in mind what the negatives of these equations
Occasionally, the ACT will test your knowledge of parabolas,
circles, or ellipses. These topics do not regularly appear on the
ACT, but it still pays to prepare: if these topics do appear, getting
them right can separate you from the crowd.
A parabola is a “U”-shaped curve that can open either
upward or downward.
A parabola is the graph of a quadratic function, which,
you may recall, follows the form
The equation of a parabola gives
you quite a bit of information about the parabola.
The vertex of the parabola is
axis of symmetry of the parabola is the line
parabola opens upward if a >
0, and downward if a <
- The y-intercept
is the point (0, c).
A circle is the collection of points equidistant from
a given point, called the center of the circle. Circles are defined
by the formula:
where (h,k) is
the center of the circle, and r is
the radius. Note that when the circle is centered at the origin, h = k =
0, so the equation simplifies to:
That’s it. That’s all you need to know about circles in
coordinate geometry. Once you know and understand this equation,
you should be able to sketch a circle in its proper place on the
coordinate system if given its equation. You should also be able
to figure out the equation of a circle given a picture of its graph
with coordinates labeled.
An ellipse is a figure shaped like an oval. It looks like
a circle somebody sat on, but it is actually a good deal more complicated
than a circle, as you can see from all the jargon on the diagram
The two foci are crucial to the definition of an ellipse.
The sum of the distances from the foci to any point on the ellipse
is constant. To understand this visually, look at the figure below.
is constant for
each point on the ellipse.
The line segment containing the foci of an ellipse with
both endpoints on the ellipse is called the major axis. The endpoints
of the major axis are called the vertices. The line segment perpendicularly
bisecting the major axis with both endpoints on the ellipse is the
minor axis. The point midway between the foci is the center of the
ellipse. When you see an ellipse, you should be able to identify
where each of these components would be.
The equation of an ellipse is:
where a, b, h,
and k are constants. With respect
to this formula, remember that:
The center of the ellipse is (h,k).
length of the horizontal axis is 2a.
length of the vertical axis is 2b.
- If a
> b, the major axis is horizontal and the minor
axis is vertical; if b > a, the major
axis is vertical and the minor axis is horizontal.