Lines
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GMAT Quantitative › Lines
Figure NOT drawn to scale.
Refer to the above figure.
True or false:
Statement 1: is a right angle.
Statement 2: and
are supplementary.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Explanation
Statement 1 alone establishes by definition that , but does not establish any relationship between
and
.
By Statement 2 alone, since same-side interior angles are supplementary, , 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. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other. and
, so it can be established that
.
What is distance between and
?
Explanation
Give the slope of the line of the equation:
Explanation
Rewrite in the slope-intercept form :
The slope is the coefficient of , which is
.
A line segment has its midpoint at and an endpoint at
. What are the coordinates of the other endpoint?
Explanation
Because we are given the midpoint and one of the endpoints, we know the x coordinate of the other endpoint will be the same distance away from the midpoint in the x direction, and the y coordinate of the other endpoint will be the same distance away from the midpoint in the y direction. Given two endpoints of the form:
The midpoint of these two endpoints has the coordinates:
Plugging in values for the given midpoint and one of the endpoints, which we can see is because it lies to the right of the midpoint, we can solve for the other endpoint as follows:
So the other endpoint has the coordinates
Determine the equation of the tangent line to the following curve at the point :
Explanation
First find the slope of the tangent line by taking the derivative of the function and plugging in the x value of the given point to find the slope of the curve at that location:
So the slope of the tangent line to the curve at the given point is . The next step is to plug this slope into the formula for a line, along with the coordinates of the given point, to solve for the value of the y intercept of the tangent line:
We now know the slope and y intercept of the tangent line, so we can write its equation as follows:
Fill in the circle with a number so that the graph of the resulting equation has slope :
None of the other responses is correct.
Explanation
Let be that missing coefficient. Then the equation can be rewritten as
Put the equation in slope-intercept form:
The coefficient of is the slope, so solve for
in the equation
A line segement on the coordinate plane has endpoints and
. Which of the following expressions is equal to the length of the segment?
Explanation
Apply the distance formula, setting
:
Examine these two equations.
Write a number in the box so that the lines of the two equations will have the same slope.
Explanation
Write the first equation in slope-intercept form:
The coefficient of , which here is
, is the slope of the line.
Now, let be the nuimber in the box, and rewrite the second equation as
Write in slope-intercept form:
The slope is , which is set to
:
Fill in the circle with a number so that the graph of the resulting equation is a horizontal line:
The graph cannot be a horizontal line no matter what number is written.
is the only number that works.
is the only number that works.
is the only number that works.
The graph is a horizontal line no matter what number is written.
Explanation
The equation of a horizontal line takes the form for some value of
. Regardless of what is written, the equation cannot take this form.
Refer to the above figure. . True or false:
Statement 1:
Statement 2: and
are supplementary.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Explanation
If transversal crosses two parallel lines
and
, then same-side interior angles are supplementary, so
and
are supplementary angles. Also, corresponding angles are congruent, so
.
By Statement 1 alone, angles and
are congruent as well as supplementary; by Statement 2 alone,
and
are also supplementary as well as congruent. Two angles that are both supplementary and congruent are both right angles, so from either statement alone,
and
intersect at right angles, so, consequently,
.