30
—
60
—
90 RIGHT TRIANGLES
In this type of right triangle, a different pattern occurs. Begin with the smallest side of the triangle, which is
the side opposite the 30-degree angle. The smallest side multiplied by
3 is equal to the side opposite the
60-degree angle. The smallest side doubled is equal to the longest side, which is the hypotenuse. For exam-
ple, if the measure of the hypotenuse is 8, then the measure of the smaller leg is 4 and the larger leg is 4
3
Pythagorean Triples
Another pattern that will help with right-triangle questions is Pythagorean triples. These are sets of whole
numbers that always satisfy the Pythagorean theorem. Here are some examples those numbers:
3
—
4
—
5
5
—
12
—
13
8
—
15
—
17
7
—
24
—
25
Multiples of these numbers will also work. For example, since 32+ 42= 52, then each number doubled
(6
—
8
—
10) or each number tripled (9
—
12
—
15) also forms Pythagorean triples.
Quadrilaterals
A quadrilateral is a four-sided polygon. You should be familiar with a few special quadrilaterals.
Parallelogram
This is a quadrilateral where both pairs of opposite sides are parallel. In addition, the opposite
sides are equal, the opposite angles are equal, and the diagonals bisect each other.
30°
60°
8
4
√¯¯¯
3
4
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Rectangle
This is a parallelogram with right angles. In addition, the diagonals are equal in length.
Rhombus
This is a parallelogram with four equal sides. In addition, the diagonals are perpendicular to each
other.
Square
This is a parallelogram with four right angles and four equal sides. In addition, the diagonals are
perpendicular and equal to each other.
Circles
■Circles are typically named by their center point. This circle is circle C.
F
G
A
CB
D
C
40°
E
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■The distance from the center to a point on the circle is called the radius,or r. The radii in this figure are
CA, CE, and CB.
■A line segment that has both endpoints on the circle is called a chord. In the figure, the chords are
.
■A chord that passes through the center is called the diameter,or d. The length of the diameter is twice
the length of the radius. The diameter in the previous figure is .
■A line that passes through the circle at one point only is called a tangent. The tangent here is line FG.
■A line that passes through the circle in two places is called a secant. The secant in this figure is line CD.
■A central angle is an angle whose vertex is the center of the circle. In this figure, ∠ACB,∠ACE, and
∠BCE are all central angles. (Remember, to name an angle using three points, the middle letter must be
the vertex of the angle.)
■The set of points on a circle determined by two given points is called an arc. The measure of an arc is
the same as the corresponding central angle. Since the m ∠ACB = 40 in this figure, then the measure of
arc AB is 40 degrees.
■A sector of the circle is the area of the part of the circle bordered by two radii and an arc (this area may
resemble a slice of pie). To find the area of a sector, use the formula , where xis the degrees of
the central angle of the sector and r is the radius of the circle. For example, in this figure, the area of the
sector formed by ∠ACB would be =
=
=
■Concentric circles are circles that have the same center.
Measurement and Geometry
Here is a list of some of the common formulas used on the GMAT exam:
A
4
1
9×36
460
360 ×62
x
360 ×62
BE
BE and CD
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■The perimeter is the distance around an object.
Rectangle P= 2l + 2w
Square P= 4s
■The circumference is the distance around a circle.
Circle C= d
■Area refers to the amount of space inside a two-dimensional figure.
Parallelogram A= bh
Triangle A=
1
2
bh
Trapezoid A=
1
2
h(b1+ b2), where b1and b2are the two parallel bases
Circle A= πr2
■The volume is the amount of space inside a three-dimensional figure.
General formula V= Bh,where B is the area of the base of the figure and his the height
of the figure
Cube V= e3,where eis an edge of the cube
Rectangular prism V= lwh
Cylinder V= πr2h
■The surface area is the sum of the areas of each face of a three-dimensional figure.
Cube SA = 6e2,where eis an edge of the cube
Rectangular solid SA = 2(lw) + 2 (lh) + 2(wh)
Cylinder SA = 2πr2+ dh
Circle Equations
The following is the equation of a circle with a radius of rand center at (h,k):
The following is the equation of a circle with a radius of r and center at (0, 0):
x2y2r2
1xh2
21yk2
2r2
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The following bullets summarize some of the major points discussed in the lessons and highlight critical
things to remember while preparing for the Quantitative section. Use these tips to help focus your review as
you work through the practice questions.
■When multiplying or dividing an even number of negatives, the result is positive, but if the number of
negatives is odd, the result is negative.
■In questions that use a unit of measurement (such as meters, pounds, and so on), be sure that all neces-
sary conversions have taken place and that your answer also has the correct unit.
■Memorize frequently used decimal, percent, and fractional equivalents so that you will recognize them
quickly on the test.
■Any number multiplied by zero is equal to zero.
■A number raised to the zero power is equal to one.
■Remember that division by zero is undefined.
■For complicated algebra questions, substitute or plug in numbers to try to find an answer choice that is
reasonable.
CHAPTER Tips and
Strategies
for the
Quantitative
Section
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