Lines - GMAT Quantitative

Given the other endpoint, you can use the distance formula to find the length of the segment:

Given the midpoint, you can use the distance formula to find the distance from the first endpoint to the midpoint, then double that to get the length of the segment:

The total length is twice that, or 10.

The answer is that either statement alone is sufficient to answer the question.

If you know only that , then you know that and , but you still need , or a way finding it.

If you know only that , you still know only that , but you don't know their actual lengths.

If you know both facts, then you know

Statement I gives us a point.

Statement II gives us the length of the segment.

We are asked to find the coordinates of the other end of the segment. However, we will need more information. Even with all of our information, we have no clue as to the orientation of the line. It could be 14 units straight up and down, it could be a perfectly horizontal line, or something inbetween, thus our answer is:

Neither I nor II are sufficient to answer the question. More information is needed.

To find the length of a segment, use the distance formula. The distance formula is given by the following:

Where your 's and 's correspond to the coordinates of the endpoints.

To find the length of Segment YZ, we need the endpoints.

Statement I gives you Point Y's coordinates.

Statement II relates Point Z's coordinates to Point Y's coordinates. Thus, we can find the point Z using Statement II.

Therefore, we need both.

Recap:

Find the length of Segment YZ

I) Point Y is located at the point .

II) Point Z has a y-coordinate twice that of Point Y, and an x-coordinate one-third of Point Y.

Use Statement II along with Statement I to find the coordinates of Point Z:

Then, use distance formula to find the length of Segment YZ:

To find the length of a segment, we need both endpoints.

Statement I gives us one endpoint.

Statement II relates and , allowing us to find the second endpoint.

Thus, we need both. Once we have both endpoints, distance is easily calculated via the distance formula or the Pythagorean theorem.

Using Statement II, we find the second endpoint to be . Use the distance formula to find your answer:

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.

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.

In order to calculate the equation of a line that passes through a point, we need one of two pieces of information. If we know another point, we can calculate the slope and solve for the -intercept, giving us the equation of the line. Alternatively, if we know the slope (which we can conclude from the parallel line in statement 2) we can calculate the -intercept and determine the equation of the line.

To find the equation of a parallel line, we need the slope and the y-intercept.

Parallel lines have the same slope, so we have that.

I and II each give us a point on the graph, so we could find the equation of the line through either of them.