GMAT Math : Geometry
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Example Questions
Example Question #661 : Geometry
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: It includes the origin.
Statement 2: It passes through Quadrant II.
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 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Infinitely many lines pass through the origin, and infinitely many lines pass through each quadrant, so neither statement alone is sufficient to answer the question.
Suppose that both statements are known to be true. Since the line passes through quadrant II, it passes through a point 
which is a negative slope.
Example Question #662 : Geometry
A line is on the coordinate plane. What is its slope?
Statement 1: The line is parallel to the line of the equation 
Statement 2: The line is perpendicular to the line of the equation 
Statement 1 ALONE is sufficient to answer the question, but Statement 2 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.
EITHER 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 insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
If Statement 1 alone holds - that is, if it is known only that the line is parallel to the line of 
If Statement 2 alone holds - that is, if it is known only that the line is perpendicular to the line of the equation 
Either statement alone will yield an answer.
Example Question #663 : Geometry
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: The line contains points in both Quadrant I and Quadrant II.
Statement 2: The line contains points in both Quadrant I and Quadrant III.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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.
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.
Examine the diagram below.
It can be seen from the red lines that no conclusions about the sign of the slope of a line can be drawn from Statement 1, since lines of positive, negative, and zero slope can contain points in both Quadrant I and Quadrant II.
If a line contains a point in Quadrant I and a point in Quadrant III, then it contains a point with positive coordinates 
which is a positive slope.
Therefore, Statement 2 alone, but not Statement 1 alone, provides a definitive answer.
Example Question #664 : Geometry
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: It passes through the point 
Statement 2: It passes through Quadrant III.
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.
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.
BOTH statements TOGETHER are insufficient to answer the question.
Examine the diagram below.
Both the red line and the green line fit the descriptions in both statements. The red line has negative slope and the green line has positive slope.
The two statements together give insufficient infomation.
Example Question #665 : Geometry
Quinn was challenged by his teacher to fill in the square, the triangle, and the circle in the diagram below with three numbers to form the equation of a line with slope 
Did Quinn succeed?
Statement 1: Quinn wrote a 3 in the square.
Statement 2: Quinn wrote a 
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.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
The equation is in standard form, with shapes replacing coefficients; that is, the equation is
Rewrite the equation in slope-intercept form:
The slope of the line of the equation is 
Example Question #56 : Lines
Ava was challenged by her teacher to fill in the triangle, the square, and the circle in the diagram below with three numbers to form the equation of a line with slope 1.
Did Ava succeed?
Statement 1: Ava wrote a 1 in the circle.
Statement 2: Ava wrote the same positive number in both the triangle and the square.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER 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.
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.
Replacing the shapes with variables, the template becomes
Divide by 
The slope is 
The slope is the ratio of the number in the square to the number in the triangle, so the number in the circle is irrelevant, making Statement 1 unhelpful.
Assume Statement 2 alone. Since the numbers in the square and the triangle are equal, 
Example Question #663 : Geometry
John's teacher gave him two equations, each with one coefficient missing, as follows:
John was challenged to write one number in each shape in order to form two equations whose lines have the same slope. Did he succeed?
Statement 1: The number John wrote in the box is three times the number he wrote in the circle.
Statement 2: John wrote 
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
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 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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.
Rewrite the two equations with variables rather than shapes:.
The first equation can be rewritten in slope-intercept form:
Its line has slope is 
is already in slope-intercept form; its line has slope 
Statement 2 alone gives the number in the circle but provides no clue to the number in the square.
Now assume Statement 1 alone. Then 
the slope of the second line. Statement 1 provides sufficient proof that John was successful.
Example Question #666 : Geometry
Does a line on the coordinate plane have undefined slope?
Statement 1: It has 
Statement 2: It passes through Quadrant II.
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 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.
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.
A line with undefined slope is a vertical line.
Infinitely many lines, some vertical and some not, pass through 
Now assume both statements are true. Then, since the line passes through Quadrant II, it passes through at least one point with negative 
a negative slope. Therefore, the slope is not undefined.
Example Question #781 : Data Sufficiency Questions
Tim was challenged by his teacher to fill in the square, the triangle, and the circle in the diagram below with three numbers to form the equation of a line with slope 
Did Tim succeed?
Statement 1: Tim wrote 
Statement 2: Tim wrote 
Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Assume both statements are true. The equation takes the form
where 
Put the equation in slope-intercept form:
The coefficient of 
Example Question #665 : Geometry
Marisol was challenged by her teacher to fill in the square and the circle in the diagram below with two numbers to form the equation of a line with slope 
Did Marisol succeed?
Statement 1: Marisol wrote a 
Statement 2: Marisol wrote a 
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.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
The pattern shows a linear equation in slope-intercept form, 
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