Coordinate Geometry - GMAT Quantitative

The slope of the first line, in terms of , is

The slope of the second line is

The slopes of two perpendicular lines have product , so we set up this equation and solve for :

or

We can eliminate the choice immediately since the slopes of two perpendicular lines cannot have the same sign. We can also eliminate and undefined, , since a line with slope 0 and a line with undefined slope are perpendicular to each other, not a line of slope -1 or 1.

Of the two remaining choices, we check for the choice that includes two numbers whose product is -1.

and

so is the correct choice.

In order for one line to be perpendicular to another, its slope must be the negative reciprocal of that line's slope. That is, the slope of any perpendicular line must be opposite in sign and the inverse of the slope of the line to which it is perpendicular:

In the given line we can see that , so the slope of any line perpendicular to it will be:

There is only one answer choice with this slope, so we know the following line is perpendicular to the line given in the problem:

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

Add the corresponding points together and divide both values by 2:

($\frac{2+4}{2}$,$\frac{9+3}{2}$) = (3, 6)

Add the x-values and divide by 2, and then add the y-values and divide by 2. Be careful of the negatives!

The midpoint of the line segment with endpoints and is , or

If , then the -coordinate is negative and the -coordinate is positive, so the midpoint is in Quadrant II. If , the reverse is true, so the midpoint is in Quadrant IV.

The midpoint of a line segment with endpoints is

.

Substitute the coordinates of the endpoints, then set each equation to the appropriate midpoint coordinate.

-coordinate:

-coordinate:

Simplify each, then solve the system of linear equations in two variables:

The two linear equations turn out to be equivalent, meaning that there are infinitely many solutions to the system. Therefore, insufficient information is given to answer the question.

Midpoint formula is as follows:

Plug in and calculate:

Midpoints can be found using the following:

Plug in our points (-6,8) and (4,26) to find the midpoint.

The midpoint formula is

Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, rewrite the equation in slope intercept form :

Slope of given line:

Negative reciprocal:

The graph of for any real number is a horizontal line. A line parallel to it is a vertical line, which has a slope that is undefined.

Rewrite in slope-intercept form:

The slope of the line is the coefficient of , which is . A line perpendicular to this has as its slope the opposite of the reciprocal of :

To find the slope of a line perpendicular to a given line, simply take the opposite reciprocal of the slope of the given line.

Since f(x) is given in slope intercept form,

.

Therefore our original slope is

So our new slope becomes:

The equation for a line in standard form is written as follows:

Where is the slope of the line and is the y intercept. By definition, the slope of a line is the negative reciprocal of the slope of the line to which it is perpendicular. So if the given line has a slope of , the slope of any line perpendicular to it will have the negative reciprocal of that slope. This gives us:

Assume Statement 1 alone. In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. The -intercept of the line of the equation can be found by substituting and solving for :

The -intercept of the line is at the origin, . It follows that the -intercept is also at the origin. Therefore, Statement 1 only gives one point on the line, and its slope cannot be determined.

Assume Statement 2 alone. The slope of the line of the equation can be calculated by putting it in slope-intercept form :

The slope of this line is the coefficient of , which is . A line perpendicular to this one has as its slope the opposite of the reciprocal of , which is

.

The question is answered.

Set in the equation:

The -intercept is .