All GMAT Math Resources
Example Questions
Example Question #11 : Perpendicular Lines
Consider the lines of the equations
and
Are these two lines parallel, perpendicular, or neither?
Statement 1:
Statement 2:
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.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.
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.
Example Question #12 : Perpendicular Lines
You are given two lines. Are they perpendicular?
Statement 1: The product of their slopes is 1.
Statement 2: The sum of their slopes is .
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient 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.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER 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.
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.
Example Question #13 : Perpendicular Lines
Are Line 1 and Line 2 on the coordinate plane perpendicular?
Statement 1: Line 1 is the line of the equation .
Statement 2: Line 2 has no -intercept.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is 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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer 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.
Example Question #104 : Coordinate Geometry
Are linear equations and perpendicular?
I) pass through the points and .
II) passes through the point and has a -intercept of .
Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.
Neither statement is sufficient to answer the question. More information is needed.
Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.
Both statements are needed to answer the question.
Either statement is sufficient to answer the question.
Both statements are needed to 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.