Perpendicular Lines - GMAT Quantitative Reasoning

To find the equation of a perpendicular line you need the slope of the line and a point on the line. We can find the slope by knowing g(x).

I) Gives us a point on h(x).

II) Gives us the y-intercept of h(x).

Either of these will be sufficient to find the rest of our equation.

Statement 1: Since the line we're looking for is perpendicular to line XY, our slope will be the inverse of line XY's slope .

The slope of our line is then . Just knowing the slope however, is not sufficient information to answer the question.

Statement 2: We're provided with a point which will allow us the write the equation.

Statement 1: We're given the equation to line AB which contains the slope. Because the line we're being asked for is perpendicular to it, we know the slope will be its inverse.

The slope of our line is then

Statement 2: We can write the equation to the perpendicular line only if we have a point that falls within that line. Luckily, we're given such a point in statement 2.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Find the equation of the line perpendicular to r(x)

I) r(x) has a slope of -15

II) The line must pass through the point (9, 96)

Recall that perpendicular lines have opposite reciprocal slopes.

Use I) to find the slope of our new line

Use II) along with our slope to find the y-intercept of our new line.

Therefore both statements are needed.

Recall that perpendicular lines have opposite, reciprocal slopes. We can find the slope of from the question.

Statement I gives us a point on , which we can use to find the y-intercept of , and then the equation.

The slope of must be the opposite reciprocal of , this makes our slope .

Statement I tells us that passes through the point , so we can use slope-intercept form to find our equation:

So, our equation is

Statement II gives us a point on , which does not help us in the slightest with . Therefore, only Statement I is sufficient.

Assume Statement 1 alone. The -intercept of the line of the equation can be found by substituting and solving for :

The -intercept of the line is the origin ; it follows that this is also the -intercept.

Therefore, Statement 1 alone yields only one point of the line, from which its equation 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

.

However, there are infintely many lines with this slope, so no further information can be determined.

Now assume both statements to be true. From Statement 1, the slope of the line is , and from Statement 2, the -coordinate of the -initercept is . Substitute in the slope-intercept form:

We are given a line, f(x), and asked to find the slope of another line, h(x).

I) Gives a point on h(x). We could plug in the point and solve for our slope. When we do this since x=0 we are unable to find the value for our slope. Therefore, statement I is not sufficient to solve the question.

II) Tells us the two lines are perpendicular. Take the opposite reciprocal of the slope of f(x) to find the slope of h(x).

Therefore,

and thus the slope of h(x) will be,

.

Statement II is sufficient to answer the question.

Statement 1: We can use the points provided to find the slope of line AB.

Since the slope we're being asked for is of a line perpendicular to line AB, their slopes are inverses of each other.

The slope of our line is then

Statement 2: Since we're provided with the line's equation, we just need to look for the slope.

Where is the slope and is the y-intercept.

In this case, we have so . Because our line is perpendicular to line AB, the slope we're looking for is

Find the slope of a line perpendicular to g(t)

I) g(t) passes through the points (9,6) and (4,-13)

II) g(t) does not pass through the origin

Perpendicular lines have opposite reciprocal slopes. For instance: a line with a slope of would be perpendicular to a line with slope of .

To find the slope of a line, we just need two points.

I) Gives us two points on g(t). We could find the slope of g(t) and then the slope of any line perpendicular to g(t).

So the slope of a line perpendicular to g(t) is equal to:

II) Is irrelevant or at least not helpful.

The lines of the two equations must have slopes that are the opposites of each others reciprocals.

Write each equation in slope-intercept form:

As can be seen, knowing the value of is necessary and sufficient to answer the question. The value of is irrelevant.

The answer is that Statement 1 alone is sufficient to answer the question, but Statement 2 alone is not sufficient to answer the question.

To determine if two lines are perpendicular, only the slope needs to be considered. The slopes of perpendicular lines are the negative reciprocals of each other. Knowing a single point on the line is not sufficient, as an infinite number of lines can pass through and individual point.

Statement 1 alone establishes by definition that , but does not establish any relationship between and .

By Statement 2 alone, since alternating interior angles are congruent, , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. By Statement 2, . and are corresponding angles formed by a transversal across parallel lines, so . is not a right angle, so .

Statement 1 alone establishes nothing about the angle makes with , as it is not part of the triangle. Statement 2 alone only establishes that .

Assume both statements are true. Then is an altitude of an equilateral triangle, making it - and - perpendicular with the base - and .

Assume Statement 1 alone. Then, as a consequence of congruence, and are congruent. They form a linear pair of angles, so they are also supplementary. Two angles that are both congruent and supplementary must be right angles, so .

Assume Statement 2 alone. Then , but without any other information about the angles that or make with , it cannot be determined whether or not.

Assume Statement 1 alone. The measure of one of the angles formed is

degrees.

Assume Statement 2 alone.

By substituting for , one angle measure becomes

The marked angles are a linear pair and thus their angle measures add up to 180 degrees; therefore, we can set up an equation:

or

yields illegal angle measures - for example,

yields angle measures for both angles; the angles are right and the lines are perpendicular.

Since the two equations are in slope-intercept form, coefficients and are the slopes of the two lines.

If , then this tells us that one of slopes and is positive and one is negative; this only eliminates the possibility of the lines being parallel.

If - or, equivalently, , then each of the slopes and is the opposite of the reciprocal of the other. This makes the lines perpendicular.

The product of the slopes of two perpendicular lines is , so from Statement 1 alone, from the fact that this product is 1, you can deduce that the slopes are not perpendicular.

Statement 2 is neither necessary nor helpful, since the sum of the slopes is irrelevant to the question.

We need to know the slopes of both lines to answer this question. Each statement gives information about only one line, so neither alone gives a definite answer.

Statement 1 tells us that Line 1 is vertical, since it is a line of the equation , for some real . Statement 2 tells us that the line, not crossing the -axis, must be parallel to the -axis and, subsequently, horizontal. A vertical line and a horizontal line are perpendicular, so the two statements together answer the question.

To find whether these two functions are perpendicular we need to find each of their slopes.

Perpendicular lines have opposite, reciprocal slopes.

Use I) to find the slope of

Use II) to find the slope of

These are not opposite reciprocals, so and are not perpendicular.