y-coordinate is its vertical position on the plane. You denote the coordinates of a point
with (x,y), where x is the point’s x-coordinate and y is the point’s y-coordinate.
The center of the coordinate plane—the intersection of the x- and y-axes—is called the
origin. The coordinates of the origin are (0,0). Any point along the x-axis has a
y-coordinate of 0 (x,0), and any point along the y-axis has an x-coordinate of 0 (0,y).
The coordinate signs (positive or negative) of points lying in the four Quadrants I–IV
in this next figure are as follows:
Notice that we’ve plotted three different points on this plane. Each point has its own
unique coordinates. (Before you continue, make sure you understand why each point
is identified by two coordinates.)
Defining a Line on the XY-Plane
You can define any line on the coordinate plane by the equation:
y 5 mx 1 b
In this equation:
• The variable m is the slope of the line.
• The variable b is the line’s y-intercept (where the line crosses the y-axis).
• The variables x and y are the coordinates of any point on the line. Any (x,y) pair
defining a point on the line can substitute for the variables x and y.
Determining a line’s slope is often crucial to solving GRE coordinate geometry
problems. Think of the slope of a line as a fraction in which the numerator indicates
the vertical change from one point to another on the line (moving left to right)
corresponding to a given horizontal change, which the fraction’s denominator indi-
cates. The common term used for this fraction is “rise over run.”
You can determine the slope of a line from any two pairs of (x,y) coordinates. In
general, if (x
1
,y
1
) and (x
2
,y
2
) lie on the same line, calculate the line’s slope as follows
(notice that you can subtract either pair from the other):
Chapter 11: Math Review: Geometry 293
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slope ~m!5
y
2
2 y
1
x
2
2 x
1
or
y
1
2 y
2
x
1
2 x
2
In applying the preceding formula, be sure to subtract corresponding values. For
example, a careless test taker calculating the slope might subtract y
1
from y
2
but
subtract x
2
from x
1
. Also, be sure to calculate “rise-over-run,” and not “run-over-rise”—
another relatively common error.
As another example, here are two ways to calculate the slope of the line defined by the
two points P(2,1) and Q(23,4):
slope ~m!5
4 2 1
23 2 2
5
3
25
slope ~m!5
1 2 4
2 2~23!
5
23
5
A GRE question might ask you to identify the slope of a line defined by a given
equation, in which case you simply put the equation in the standard form y 5 mx 1 b,
then identify the m-term. Or, it might ask you to determine the equation of a line, or
just the line’s slope (m)ory-intercept (b), given the coordinates of two points on the
line.
29. On the xy-plane, at what point along the vertical axis (the y-axis) does the
line passing through points (5, 22) and (3,4) intersect that axis?
(A) 28
(B)
2
5
2
(C) 3
(D) 7
(E) 13
The correct answer is (E). The question asks for the line’s y-intercept (the
value of b in the general equation y 5 mx 1 b). First, determine the line’s slope:
slope m 5
y
2
2 y
1
x
2
2 x
1
5
4 2~22!
3 2 5
5
6
22
523
In the general equation (y 5 mx 1 b), m 523. To find the value of b,
substitute either (x,y) value pair for x and y, then solve for b. Substituting
the (x,y) pair (3,4):
y 523x 1 b
4 523~3!1b
4 529 1 b
13 5 b
PART IV: Quantitative Reasoning294
ALERT!
On the xy -plane, a line’s slope
is its “ rise over run”—the
vertical distance between two
points divided by the
horizontal distance between
the same two points. When
finding a s lope, be careful not
to calculate “run over rise”
instead!
www.petersons.com
To determine the point at which two nonparallel lines intersect on the coordinate
plane, first determine the equation for each line. Then, solve for x and y by either
substitution or addition-subtraction.
30. In the standard xy-coordinate plane, the xy-pairs (0,2) and (2,0) define a
line, and the xy-pairs (22,21) and (2,1) define another line. At which of
the following points do the two lines intersect?
(A)
S
4
3
,
2
3
D
(B)
S
3
2
,
4
3
D
(C)
S
2
1
2
,
3
2
D
(D)
S
3
4
, 2
2
3
D
(E)
S
2
3
4
, 2
2
3
D
The correct answer is (A). Formulate the equation for each line by
determining slope (m), then y-intercept (b). For the pairs (0,2) and (2,0):
y 5
S
0 2 2
2 2 0
D
x 1 b ~slope 521!
0 522 1 b
2 5 b
The equation for the line is y 52x 1 2. For the pairs (22, 21) and (2,1):
y 5
S
1 2~21!
2 2~22!
D
x 1 b
S
slope 5
1
2
D
1 5
1
2
~2!1b
0 5 b
The equation for the line is y 5
1
2
x. To find the point of intersection, solve for
x and y by substitution. For example:
1
2
x
52x 1 2
3
2
x
5 2
x 5
4
3
y 5
2
3
The point of intersection is defined by the coordinate pair
S
4
3
,
2
3
D
.
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Graphing a Line on the XY-Plane
You can graph a line on the coordinate plane if you know the coordinates of any two
points on the line. Just plot the two points and then draw a line connecting them. You
can also graph a line from one point on the line, if you know either the line’s slope or
its y-intercept.
A GRE question might ask you to recognize the value of a line’s slope (m) based on a
graph of the line. If the graph identifies the precise coordinates of two points, you can
determine the line’s precise slope and the entire equation of the line. Even without
any precise coordinates, you can still estimate the line’s slope based on its appearance.
Lines That Slope Upward from Left to Right:
• A line sloping upward from left to right has a positive slope (m).
• A line with a slope of 1 slopes upward from left to right at a 45° angle in relation
to the x-axis.
• A line with a fractional slope between 0 and 1 slopes upward from left to right but
at less than a 45° angle in relation to the x-axis.
• A line with a slope greater than 1 slopes upward from left to right at more than a
45° angle in relation to the x-axis.
Lines That Slope Downward from Left to Right:
• A line sloping downward from left to right has a negative slope (m).
• A line with a slope of 21 slopes downward from left to right at a 45° angle in
relation to the x-axis.
• A line with a fractional slope between 0 and 21 slopes downward from left to
right but at less than a 45° angle in relation to the x-axis.
• A line with a slope less than 21 (for example, 22) slopes downward from left to
right at more than a 45° angle in relation to the x-axis.
PART IV: Quantitative Reasoning296
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Horizontal and Vertical Lines:
• A horizontal line has a slope of zero (0) (m 5 0, and mx 5 0).
• A vertical line has either an undefined or an indeterminate slope (the fraction’s
denominator is zero (0)).
Chapter 11: Math Review: Geometry 297
TIP
Parallel lines have the same
slope (the same m-term in the
general equation). The slope
of a line perpendicular to
another is the negative
reciprocal of the other line’s
slope. (The product of the two
slopes is 21.)
www.petersons.com
31.
P
Referring to the xy-plane above, which of the following could be the
equation of line P?
(A) y 5
2
5
x 2
5
2
(B) y 52
5
2
x 1
5
2
(C) y 5
5
2
x 2
5
2
(D) y 5
2
5
x 1
2
5
(E) y 52
5
2
x 2
5
2
The correct answer is (E). Notice that line P slopes downward from left
to right at an angle greater than 45°. Thus, the line’s slope (m in the
equation y 5 mx 1 b) ,21. Also notice that line P crosses the y-axisata
negative y-value (that is, below the x-axis). That is, the line’s y-intercept (b
in the equation y 5 mx 1 b) is negative. Only choice (E) provides an
equation that meets both conditions.
Midpoint and Distance Formulas
To be ready for GRE coordinate geometry, you’ll need to know midpoint and distance
formulas. To find the coordinates of the midpoint of a line segment, simply average
the two endpoints’ x-values and y-values:
x
M
5
x
1
1 x
2
2
and y
M
5
y
1
1 y
2
2
For example, the midpoint between (23,1) and (2,4) 5
S
23 1 2
2
,
1 1 4
2
D
or
S
2
1
2
,
5
2
D
.
A GRE question might simply ask you to find the midpoint between two given points,
or it might provide the midpoint and one endpoint and then ask you to determine the
other point.
32. In the standard xy-coordinate plane, the point M(21,3) is the midpoint of a
line segment whose endpoints are A(2,24) and B. What are the xy-
coordinates of point B?
(A) (21,22)
(B) (23,8)
(C) (8,24)
(D) (5,12)
(E) (24,10)
PART IV: Quantitative Reasoning298
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The correct answer is (E). Apply the midpoint formula to find the x-coordinate
of point B:
21 5
x 1 2
2
22 5 x 1 2
24 5 x
Apply the midpoint formula to find the y-coordinate of point B:
3 5
y 2 4
2
6 5 y 2 4
10 5 y
To find the distance between two points that have the same x-coordinate (or y-coor-
dinate), simply compute the difference between the two y-values (or x-values). But if
the line segment is neither vertical nor horizontal, you’ll need to apply the distance
formula, which is actually the Pythagorean theorem in thin disguise (it measures the
length of a right triangle’s hypotenuse):
d 5
=
~x
1
2 x
2
!
2
1~y
1
2 y
2
!
2
For example, the distance between (23,1) and (2,4) 5
=
~23 2 2!
2
1~1 2 4!
2
5
=
25 1 9 5
=
34.
A GRE question might ask for the distance between two defined points, as in the
example above. Or it might provide the distance, and then ask for the value of a
missing coordinate—in which case you solve for the missing x-value or y-value in the
formula.
Figures in Two Dimensions
Up to this point in the chapter, the coordinate geometry tasks you’ve learned to
perform have all involved points and lines (line segments) only. In this section, you’ll
explore coordinate-geometry problems involving two-dimensional geometric figures,
especially triangles and circles.
Triangles and the Coordinate Plane
On the GRE, a question might ask you to find the perimeter or area of a triangle
defined by three particular points. As you know, either calculation requires that you
know certain information about the lengths of the triangle’s sides. Apply the distance
formula (or the standard form of the Pythagorean theorem) to solve these problems.
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33. On the xy-plane, what is the perimeter of a triangle with vertices at points
A(21,23), B(3,2), and C(3,23)?
(A) 12
(B)
10 1 2
=
3
(C)
7 1 5
=
2
(D) 15
(E
9 1
=
41
The correct answer is (E). The figure below shows the triangle on the coor-
dinate plane:
B
C
A
AC 5 4 and BC 5 5. Calculate AB (the triangle’s hypotenuse) by the
distance formula or, since the triangle is right, by the standard form of the
Pythagorean theorem: (AB
)
2
5 4
2
1 5
2
;(AB)
2
5 41; AB 5
=
41. The
triangle’s perimeter 5 4 1 5 1
=
41 5 9 1
=
41.
Note that, since the triangle is right, had the preceding question asked for
the triangle’s area instead of perimeter, all you’d need to know are the
lengths of the two legs (
AC and BC). The area is
S
1
2
D
~4!~5!5 10.
To complicate these questions, the test makers might provide vertices that do not
connect to form a right triangle. Answering this type of question requires the extra
step of finding the triangle’s altitude. Or they might provide only two points, then
require that you construct a triangle to meet certain conditions.
PART IV: Quantitative Reasoning300
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34. On the xy-plane, the xy-coordinate pairs (26,2) and (214,24) define one
line, and the xy-coordinate pairs (212,1) and (23,211) define another line.
What is the unit length of the longest side of a triangle formed by the
y-axis and these two lines?
(A) 15
(B) 17.5
(C) 19
(D) 21.5
(E) 23
The correct answer is (D). For each line, formulate its equation by deter-
mining slope (m), then y-intercept (b).
For the Pairs (26,2) and
(214,24)
For the Pairs (212,1) and
(23,211)
y 5
6
8
x 1 b
S
slope 5
3
4
D
2 5
3
4
~26!1b
2 5
24
1
2
1 b
2 1 4
1
2
5 b
6
1
2
5 b
y 5
212
9
x 1 b
S
slope 52
4
3
D
1 5
2
4
3
~212!1b
1 5
48
3
1 b
1 2 16 5 b
215 5 b
The two y-intercepts are 6
1
2
and 215. Thus the length of the triangle’s side along
the y-axis is 21.5. But is this the longest side? Yes. Notice that the slopes of the
other two lines (l
1
and l
2
in the next figure) are negative reciprocals of each other:
S
3
4
DS
2
4
3
D
521. This means that they’re perpendicular, forming the two legs of
a right triangle in which the y-axis is the hypotenuse (the longest side).
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If the preceding question had instead asked for the point at which the two lines
intersect, you would answer the question formulating the equations for both lines,
then solving for x and y with this system of two equations in two variables.
Circles and the Coordinate Plane
A GRE question might ask you to find the circumference or area of a circle defined by
a center and one point along its circumference. As you know, either calculation
requires that you know the circle’s radius. Apply the distance formula (or the
standard form of the Pythagorean theorem) to find the radius and to answer the
question.
35. On the xy-plane, a circle has center (2,21), and the point (23,3) lies along
the circle’s circumference. What is the square-unit area of the circle?
(A) 36p
(B)
81p
2
(C) 41p
(D) 48p
(E) 57p
The correct answer is (C). The circle’s radius is the distance between its center
(2,21) and any point along its circumference, including (23,3). Hence, you can
find r by applying the distance formula:
=
~23 2 2!
2
1~3 2~21!!
2
5
=
25 1 16 5
=
41
The area of the circle 5p~
=
41!
2
5 41p.
In any geometry problem involving right triangles, keep your eyes open for the
Pythagorean triplet in which you’ll see the correct ratio, but the ratio is between the
wrong two sides. For instance, in the preceding problem, the lengths of the two legs of
a triangle whose hypotenuse is the circle’s radius are 4 and 5. But the triangle does
not conform to the 3:4:5 Pythagorean side triplet. Instead, the ratio is 4:5:
=
41.
PART IV: Quantitative Reasoning302
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. the point’s y-coordinate.
The center of the coordinate plane the intersection of the x- and y-axes—is called the
origin. The coordinates of the origin are. line.
• The variable b is the line’s y-intercept (where the line crosses the y-axis).
• The variables x and y are the coordinates of any point on the line.