Perpendicular Lines - GRE Quantitative Reasoning

Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2.

To find if something is perpendicular, you need to first know the slope of your given line. Based on your points, this is easy. Recall that slope is merely:

This is:

Since a perpendicular line has a slope that is both opposite in sign and reciprocal, you need to choose a line with a slope of . The only possible option is, therefore,

Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.

The slope of this line is . The negative reciprocal will be , which will be the slope of the perpendicular line.

Now we need to find the answer choice with this slope by converting to slope-intercept form.

This equation has a slope of , and must be our answer.

switch 4x+ 3y = 6 to "y=mx+b" form

3y= -4x + 6

y = -4/3 x + 2

m = -4/3; the perpendicular line will have the negative reciprocal of this line so it would be 3/4

First, we must solve the equation for y to determine the slope: y = 2_x_ + 3/2

By looking at the coefficient in front of x, we know that the slope of this line has a value of 2. To fine the slope of any line perpendicular to this one, we take the negative reciprocal of it:

slope = m , perpendicular slope = – 1/m

slope = 2 , perpendicular slope = – 1/2

The line will be perpendicular if the slope is the negative reciprocal.

First we need to find the slope of our line between points (22,24) and (31,4). Slope = rise/run = (24 – 4)/(22 – 31) = 20/–9 = –2.22.

The negative reciprocal of this must be a positive fraction, so we can eliminate y = –3_x_ + 5 (because the slope is negative).

The negative reciprocal of –2.22, and therefore the slope of the perpendicular line, will be –1/–2.22 = .45, so we can also eliminate y = x (slope of 1).

Now let's look at the line between points (9, 5) and (48, 19). This slope = (5 – 19)/(9 – 48) = .358, which is incorrect.

The next answer choice is y = .45_x_ + 10. The slope is .45, which is what we're looking for so this is the correct answer.

To double check, the last answer choice is the line between (4, 7) and (7, 4). This slope = (7 – 4) / (4 – 7) = –1, which is also incorrect.

We first need to recall the following relationships:

Lines with the same slope and same -intercept are really the same line.

Lines with the same slope and different -intercepts are parallel.

Lines with slopes that are negative reciprocals are perpendicular.

Then we identify the slopes of the two lines by comparing the equations to the slope-intercept form , where is the slope and is the -intercept. By inspection we see the lines have slopes of and . Since these are different, the "parallel" and "same line" choices are eliminated. To test if the slopes are negative reciprocals, we take one of the slopes, change its sign, and flip it upside-down. Starting with and changing the sign gives , then flipping gives . This is the same as the slope of the second line, so the two slopes are negative reciprocals and the lines are perpendicular.

If lines are perpendicular, then their slopes will be negative reciprocals.

First, we need to find the slope of the given line.

Because we know that our given line's slope is , the slope of the line perpendicular to it must be .

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, , where m is the slope of the line. Two lines are perpindicular to each other if one slope is the negative and reciprocal of the other.

The first step of this problem is to get it into the form, , which is . Now we know that the slope, m, is . The reciprocal of that is , and the negative of that is . Therefore, any line that has a slope of will be perpindicular to the original line.

To begin, the best thing to do is to put your equation into slope-intercept format. That is, into the format:

For your equation, you need to solve for :

, which is the same as

Then, divide both sides by :

So, the slope of this line is . The perpendicular of a line is opposite and reciprocal. Therefore, the perpendicular line will have a slope of . Of the options given, only matches this (which you can figure out when you solve for ).

Remember, to be perpendicular, two lines must have opposite and reciprocal slopes. Therefore, you need to begin by solving for the slope of your given line. You do this by finding:

For two points and , this is:

For our points, this is:

The slope of the perpendicular line will be (remember) opposite and reciprocal. Therefore, it will be . Now, among your equations, the only one that has this slope is:

If you solve this for , you get:

According to the slope-intercept form (), this means that the slope is .

Perpendicular lines have opposite and reciprocal slopes. Therefore, let us find the slope of our line and appropriately modify it to find the perpendicular line. To find the slope of our line, remember that in the slope-intercept form (y = mx + b), m represents the slope.

3x + 4y = 5 => 4y = 5 - 3x => y = (-3/4)x + 5/4

Therefore, the slope of our line is -3/4. The perpendicular to this would be 4 / 3. Therefore, only y = (4/3)x - 15 is perpendicular.

To solve this, we need to use the point-slope form. However, to do this, we need to ascertain the slope of the perpendicular line. For this, get 8y + 4x = 10 into slope-intercept form:

8y = -4x + 10; y = (-1/2)x + 1.25

Therefore, the slope of this line is -1/2 and its perpendicular (opposite and reciprocal) slope is 2.

The point slope form is: (y – y1) = m * (x – x1)

For our data it is: y – 4 = 2 * (x – 5)

Simplifying, we get: y – 4 = 2 x – 10; y = 2x – 6

Rewrite the equation in form.

The perpendicular line will have a slope which is the negative reciprocal of the slope .

To find if something is perpendicular, you need to first know the slope of your given line. Based on your points, this is easy. Recall that slope is merely:

This is:

Since a perpendicular line has a slope that is both opposite in sign and reciprocal, you need to choose a line with a slope of . The only possible option is, therefore,

Find the slope of the given line. The perpendicular slope will be the opposite reciprocal of the original slope. Use the slope-intercept form (y = mx + b) and substitute in the given point and the new slope to find the intercept, b. Convert back to standard form of an equation: ax + by = c.

The question puts the line in point-slope form y – y1 = m(x – x1), where m is the slope. Therefore, the slope of the original line is 1/2. A line perpendicular to another has a slope that is the negative reciprocal of the slope of the other line. The negative reciprocal of the original line is _–_2, and is thus the slope of its perpendicular line.

The equation of a line is where is the slope.

Rearrange the equation to match this:

For the perpendicular line, the slope is the negative reciprocal;

therefore

First we must find the slope of the given line. The slope of y = –3x – 4 is –3. The slope of the perpendicular line is the negative reciprocal. This means you change the sign of the slope to its opposite: in this case to 3. Then find the reciprocal by switching the denominator and numerator to get 1/3; therefore the slope of the perpendicular line is 1/3.