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# Gruber's Complete GRE Guide 2015 Gruber, Gary R

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```COMPLETE GRE GUIDE 2015
Know and Use Facts about triangles
By remembering these facts about triangles, you can often save yourself a lot of time
and trouble.
Math
StratEg
y18
I.

a
x° y°
b
If a 5 b, then x 5 y
The base angles of an isosceles triangle are equal.

a
x° y°
b
If x 5 y, then a 5 b
If the base angles of a triangle are equal, the triangle
is isosceles.
II.
z°x°
y°

A B
D
C
1
2
A B
D
C
1
2
is a straight line.
Then, x 5 y 1 z
The measure of an exterior angle is equal to the sum of
the measures of the remote interior angles.
III.
a x° y°
b
If a , b, then y , x
a x° y°
b
If y , x, then a , b
In a triangle, the greater angle lies opposite the greater
side.
IV.
Similar Triangles

A
B C
a
bc
f e
d
D
E F
If \u394ABC ~ \u394DEF, then
m+A 5 m+D
m+B 5 m+E
m+C 5 m+F

m A m D
m B m E
m C m F
d
a
e
b
f
cand
=
=
=
= =
B B
B B
B B
V.
A
B C
m+A 1 m+B 1 m+C 5 180º
The sum of the interior angles of a triangle is 180 degrees.
VI.
A
B
D
C
#=
The area of a triangle is one-half the product of the
altitude to a side and the side.
Note: If m +A 5 90°,
Area also Area also AB AC2
#=
GRE2015_P04.indd 118 4/18/14 11:29 AM
STRATEGY SECTION \u2022 119
VII.

a c
b
x°
y°
In a right triangle,
c2 5 a2 1 b2
and x°1 y° 5 90°
VIII. Memorize the following standard triangles:

15 17
8 7
5
12
24
54
2
1
1
1
1 1
1
1
60°
60°
60° 60°
45°
45°
45°
45°
30°
3
9
4140
13
25
3
2
2
2
2
2
IX. ba
c
a 1 b . c
a 1 c . b
b 1 c . a
The sum of the lengths of two sides of a triangle is
greater than the length of the third side. (This is like
saying that the shortest distance between two points is
a straight line.)
example 1
In the diagram below, what is the value of x?
24 x
10
M
N P
(A) 20
(B) 25
(C) 26
(D) 45
(E) 48
Choice C is correct.
Method 1: Use Statement VII. Then,
x2 5 242 1 102
5 576 1 100
5 676
Method 2: Look at Statement VIII. Notice that \u394MNP is
similar to one of the standard triangles:
24
x
10
M
N P
13
12
5
This is true because
24
12
10
5
= (Look at IV).
= =, 26 ( )x or x Answer24
12 13Hence
example 2
If Masonville is 50 kilometers due north of Adamston
and Elvira is 120 kilometers due east of Adamston, then
the minimum distance between Masonville and Elvira is
(A) 125 kilometers
(B) 130 kilometers
(C) 145 kilometers
(D) 160 kilometers
(E) 170 kilometers
Choice B is correct. Draw a diagram first.
Masonville
50 km
x
The given information translates into the diagram
above. Note Statement VIII. The triangle above is a
multiple of the special 5\u201312\u201313 right triangle.
GRE2015_P04.indd 119 4/18/14 11:29 AM
120 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015
50 5 10(5)
120 5 10(12)
Thus, x 5 10(13) 5 130 kilometers
(Note: The Pythagorean Theorem could also have been
used: 502 1 1202 5 x2.)
example 3
C
b°
a° c°
B
A
(Note: Figure is not drawn to scale.)
In triangle ABC, if a . c, which of the following is true?
(A) BC 5 AC
(B) AB . BC
(C) AC . AB
(D) BC . AB
(E) BC . AC
Choice D is correct. (Remember triangle inequality facts.)
From basic geometry, Statement III, we know that, since
m+BAC . m+BCA, then leg opposite +BAC . leg
opposite +BCA, or
BC . AB
example 4
C
45°
B
A
(Note: Figure is not drawn to scale.)
The triangle above has side BC 5 10, angle B 5 45°, and
angle A 5 90°. The area of the triangle
(A) is 15
(B) is 20
(C) is 25
(D) is 30
(E) Cannot be determined.
Choice C is correct.
First find angle C using Statement V.
90° 1 45° 1 m+C 5 180°
So m+C 5 45°.
Using Statement I, we find AB 5 AC,
since m+B 5 m+C 5 45°.
Since our right triangle ABC has BC 5 10, using
Statement VIII (the right triangle , ,2
2
2
2 1), multiply
by 10 to get a right triangle:
, ,2
10 2
2
10 2 10
5AB 2
10 2 2Thus side = =
5AC 2
10 2 2side = =
Now the area of triangle ABC, according to Statement
VI, is
5 5 252
2 2
2
25 2# #= =
example 5
C
x°
x°55°
45°
D B
A
In the figure above, what is the value of x?
(A) 30
(B) 40
(C) 50
(D) 80
(E) 100
Choice B is correct.
Remember triangle facts. Use Statement II.
\u2220ADB is an exterior angle of \u394 ACD, so
m\u2220ADB 5 x 1 x 5 2x 1
In \u394ADB, the sum of its angles 5 180 (Statement V), so
m\u2220ADB 1 55 1 45 5 180
or m\u2220ADB 1 100 5 180
Equating 1 and 2 , we have
2x 5 80
GRE2015_P04.indd 120 4/18/14 11:29 AM
STRATEGY SECTION \u2022 121
When Calculating answers, Never Multiply and/or Do Long
Division If you Can reduce First
Note: On the GRE exam, because calculators are permitted, you may do the following problems
with a calculator also. But it would be wise for you to see the other approach too\u2014how the
problem can be solved without the use of a calculator.
Math
StratEg
y19
example 1
If , thenw w45 40
81 150
#
#= =
(A) 3
(B) 6 4
3
(C) 7 4
1
(D) 9
(E) 20 4
1
Do not
multiply in} 81 3 150 and 45 3 40 to getthis case.
,
,
1 800
12 150
Factor first: 9 5 4 10
9 9
45 40
81 150
15 10Factor first
# # #
# # #
? ?
S S
Then cancel like factors in numerator and denominator:
9 5 4 10
9 9 15 10
# # #
# # #
Reduce further:
5
9 5
4
3
Reduce further
#
# #
27
4
3
=
Thus, Choice B is correct.
example 2
3 3 3
4 4 4
3 3 3
2 2 2
+ +
+ + =
(A)
27
16
(B)
9
8
(C)
3
4
(D)
27
64
(E) 81
512
Choice A is correct.
example 6
5
4
a
(Note: Figure is not drawn to scale.)
Which of the following represent all of the possibilities
for the value of a in the figure above?
(A) 1 , a , 9
(B) 4 , a , 5
(C) 0 , a , 9
(D) 4 , a , 9
(E) 5 , a , 9
Choice A is correct. From Statement IX, since the sum
of the lengths of two sides of a triangle is greater than
the length of the third side, we have:
a 1 5 . 4 1
a 1 4 . 5 2
5 1 4 . a 3
From 2 we get:
a . 1.
From 3 we get:
9 . a.
This means that
9 . a . 1, or 1 , a , 9.
GRE2015_P04.indd 121 4/18/14 11:29 AM
122 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015
3 3 3
4 4 4
3 3 3
2 2 2
+ +
+ + =
Factor and reduce:
3( )
3( )
3
4
3
2
5
27
example 3
6 7 8 9 12 14 ,x x
18If then# # # # #= =
(A) 2
1
(B) 1
(C) 4
(D) 8
(E) 12
Choice B is correct.
Given: 6 7 8 9 12 14x
18
# # #
# #= 1

6 7 8
12 14x 9
18so that
# # #
# #= 2
Do not multiply the numbers out in the numerator and
denominator of 2 ! It is too much work! Rewrite 2 .
Factor and reduce:
x 5
6 7 8
12 14
6 7 8 9
2 6 2 7 2 9
2 2
8
8
9
18
8
2
# # #
# #
# # #
# # # # #
# #
=
= = =

6 7 8
12 14
6 7 8 9
2 6 2 7 2 9
2 2
8
8
9
18
8
2
# # #
# #
# # #
# # # # #
# #
=
example 4
If 21,y y27
81 then# = =
(A) 21
1
(B) 7
1
(C) 3
(D) 7
(E) 21
Choice D is correct.
: 81 21Given y27
#
=
Multiply both sides by 27 to get 81 3 y 5 21 3 27.
y 81
21 27#=
Factor and reduce:

9 9
3 7 3 9
3 3
3 3
7
y
y
7
\$
\$ # \$
\$
\$ #
=
=
example 5
Find the value of
2
7 10
2y
y y22 + rounded to the nearest
whole number if y 5 8.000001.
(A) 2
(B) 3
(C) 5
(D) 6
(E) 16
Choice B is correct.
Given: 2
7 10```