Coordinate geometry questions test your ability to interpret and deal with geometric figures on an xy-graph.

The point at which the x and y axes meet is called the origin. As you go to the right on the x-axis, you move into positive numbers, and as you go to the left, you move into negative numbers. Likewise, as you go up on the y-axis, you move into positive numbers, while moving down brings you into negative numbers (xy-graphs on the SAT will not have their intervals as clearly marked as this diagram does).

To specify a particular point on any xy-graph, you must use ordered pairs. Ordered pairs are two numbers put in parentheses and separated by commas: (x,y). The first number represents a position on the x-axis and the second a position on the y-axis. To think of it another way, the first number tells you how far the point is to the left or right of the origin, and the second number tells you how far up or down the point is.

For example, to graph (2, 3), you should move two spaces to the right (since 2 is positive) and three spaces up (since 3 is positive). To graph (–2, 1), move two spaces to the left (since –2 is negative) and one space up (since 1 is positive). To graph (1.5, –1), move 1.5 spaces to the right, and 1 space down (since –1 is negative).

By the time the SAT rolls around, you should be able to graph or interpret any ordered pairs that you might see. You should know how to calculate the distance between two pairs and be able to state what the mirror of an ordered pair might be.

The distance between two ordered pairs is easy to calculate. If you are asked to calculate the distance between (2,3) and (2,–2), the first thing you should do is figure out where each appears on the xy-graph. You may be comfortable enough to do this without actually drawing out the graph, but making a little sketch is never a bad idea.

As the ordered pairs indicate, there is no difference between the two pairs in reference to the x-axis: both points are the same number of spaces to the right. But the points are not equal in reference to the y-axis. One is positive 3, and so it is three spaces up, while the other is –2, meaning it is two spaces down. Therefore, to get from one to the other, you would have to move a total of five spaces, mathematically (3 – (–2)) = 5.

In working out this example we’ve come to a rule about finding the distance between two ordered pairs: if the pairs have one coordinate in common, then the distance between the two pairs is the absolute value or difference between the dissimilar coordinates. For example, the distance between: (4,6) and (2,6) is 4–2 = 2. The distance between the points (–2, 7) and (–2, –1) is 7 – (–1) = 8.

For the SAT, you will not have to calculate the length between two ordered pairs with different coordinates in each place. You will not, for instance, have to calculate the distance between (1,6) and (–4,2).

Points can be reflected across the x- or y-axis or through the origin. Pairs of reflected points are equidistant from each axis and from the origin. Take a look at the following problem:

Because AC = BC, this drawing tells you that point A is the same distance to the left of the y-axis as B is to the right of it. It also tells you that line AB is parallel to the x-axis, which means that the y coordinate for each ordered pair will be the same. Immediately we know that the coordinates of A are (x,2). To find x, all we need is a little common sense. If the coordinates of point B are (3,2) and A is just as far from the y-axis but on the opposite side, then the size of the x coordinate of point A must be the same as the x-coordinate for point B. Only the sign must be different. Therefore, the coordinates of A must be (–3,2).

Points that reflect through the origin are only slightly more difficult to deal with:

Here, the signs of the coordinates will be opposite, but their values will be the same. For example, in the diagram above, if point A were (–4,–1), point B would have to be (4,1).

In addition to its familiar meaning, the word “slope” has a precise mathematical definition. The slope of a line is known as the “rise over run,” or the change in y divided by the change in x. To find the slope of a line, pick any two points on that line. Then subtract their y-coordinates and their x-coordinates, in the same order, and divide the difference of the y-coordinates by the difference of the x-coordinates. For example, to find the slope of the line which passes through the points (2, 5) and (0, 1):

A slope of 2 means that for each unit x increases anywhere on the line, y increases by 2 units; whenever x decreases by 1, y decreases by 2.

The sign, positive or negative, of a slope indicates how the line moves away from the origin. Since slope is the measure of “rise over run”—the change in y divided by the change in x—the slope of a line will be positive when the change in y is positive and the change in x is positive, or when both are negative. This is true because of the division rules we covered earlier: both a positive number divided by a positive number and a negative number divided by a negative number will result in positive quotients.

Alternatively, if the change in y and the change in x have different signs, meaning one is positive and one is negative, the slope of the line will be negative.

For the SAT, you should be able to look at two sets of coordinate points located on a line, such as (2, 3) and (4, 1), and be able to tell if the slope of that line is positive or negative. In the case of these two points, you can see that the x-coordinate is increasing from 2 to 4 while the y-coordinate is decreasing from 3 to 1, meaning that the slope must be negative.

A line with positive slope on a graph will rise moving from left to right. A line with negative slope will lower moving from left to right.

For the SAT, you should be able to look at a line on a coordinate graph and know immediately whether it has a positive or negative slope.

Horizontal and vertical lines have special slope values. Horizontal lines always have the same y-coordinate. There is no rise over run.

The slope of this line is: m = ^{3 –3}/_{2 – (–3)} = ⁰/₅ = 0. The slope of any horizontal line is 0, indicating that as x increases or decreases, y does not change.

For vertical lines, x remains constant as y increases or decreases:

Vertical lines have no “run,” so the change in x is 0. Yet the slope equation demands that you divide by the change in x. Division by 0 is impossible and makes the slope undefined: m = ^{3 – (–1)}/_{–1 – (–1)} = ⁴/₀ = undefined. Since it is impossible to divide a number by zero, the slope of any vertical line is undefined.

Two lines are parallel if they have the same slope. Parallel lines, when extended, do not intersect at any point because they have the exact same rise over run, so one line can never get closer to another. In the image below, lines AB and CD have the same slope, so they are parallel.

Perpendicular lines intersect each other at right angles. Two lines are perpendicular if their slopes are negative reciprocals of each other. Perpendicular lines have exact opposite rise over runs. For example, in the image below, line AB rises three units for every four units it moves to the right, so it has a slope of ³/₄. Line CD falls four spaces for each three spaces it moves to the right, so its slope is –⁴/₃.

Not all lines have fractional slopes, of course. To find the perpendicular slope for a line with a whole number slope, for example a line with a slope of 2, just think of the whole number as if it was a fraction ² /₁ and take the negative reciprocal: –¹/₂