Strategies, Sample Questions, and Random Ramblings.
July 19, 2015
GMAT coordinate geometry is another topic that is not overly prevalent on the exam, but again worth preparing for so that you do not waste points that are attainable. So, let’s start with the basics and build into some of the concepts that will be tested on the GMAT.
Above is the coordinate plane. The two lines, x and y, are axes of the coordinate plane and are perpendicular to each other intersecting at the origin, O, which represents the center. The x-axis runs left to right and the y-axis runs up and down. These two lines divide the four quadrants of the plane as labeled above. For example, the lower left corner is called quadrant III.
You will use the coordinate plane to plot points, lines and shapes (made of line segments). The GMAT may ask you a variety of things about what is plotted on the coordinate plane, so let’s look at how points are plotted and we will build into lines and shapes.
First, it is important to understand that a point is written in this format: (x,y). The “x” represents the movement to the right or left from the origin, while the “y” represents movement up and down. The origin is located at point (0,0). The planes below show examples of how points will be plotted.
If, for example you were to plot the point (3,-2), you would move 3 units to the right and then 2 down to be in quadrant IV. That is pretty much it when it comes to plotting points.
Obviously the GMAT is going to do things a bit more complicated than just plotting points. You can use two plotted points to make a line and a common thing discussed related to lines is the slope. The slope generally measure the ‘steepness’ of a line and can be calculated using the formula:
Slope is often viewed as rise over run. The graph below shows three slopes that are all positive. The slope of b = 1, while a is greater than 1 and c is less than 1.
The basic concept to know is that as a line flattens, its slope gets closer and closer to zero, while it gets larger the steeper the line. In fact, a horizontal line has a slope of 0 and a vertical line has an undefined slope - this is because the x values are identical and you cannot divide by a zero. The same principle holds true for the lines below, except all of the slopes are negative.
The slope of b is -1, while the slope of a is less than -1 (-3 for example) and c is greater than -1 (-1/2 for example).
Let’s look at a specific example of a line. Lines follow the equation: y = mx + b , where m is the slope and b is the y-intercept, or the place where the line crosses the y-axis.
To calculate the slope of this line we would use the formula we previously discussed for the slope of a line along with the coordinates on the line:
Thus, the equation would now look like this: y = .75x + b
In order to solve for for b, all you have to do is plug one of the coordinates into the equation. Let’s use (2,1).
So the final equation ends up like this: y = .75x − .5
Although not in the equation for the line, you may be asked about the x-intercept as well, or where the line crosses the x-axis. All that needs to be done to calculate this is set y equal to zero and solve for x.
Other slope related information that you will be required to recall is that parallel lines have the same slope and perpendicular lines have slopes that are the negative reciprocal of each other. So, the line that is perpendicular to 2 is -1/2 and the line that is perpendicular to 1/4 is -4.
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