BT N14
CHAPTER 4: ADDITIONAL TOPICS
4-1. Understanding Geometric Relationships ................... 2
4-2. Distance and Area ...................................................... 4
4-3. Understanding Basic Trigonometry ........................... 7
4-4. Understanding Complex Numbers............................. 8
Understanding Geometric Relationships
1
4
Q
M
l
N
x°
m
(x + 5)°
y°
O
P
In the figure above, MNOP is a square and Q is the
20
midpoint of MN . If QO =
, what is the area of
3
square MNOP ?
Lines l and m are parallel in the figure above.
Which of the following expresses the value of y in
terms of x ?
A) 95 - 2x
B) 165 - 2x
C) 175 - 2x
2
D) 185 - 2x
5
40°
x°
y
l
87°
(12, 13)
(3, 13)
m
(18, 5)
(3, 5)
x
O
Lines l and m are parallel in the figure above. What
is the value of x ?
In the figure above, what is the distance between
the midpoints (not shown) of the two line
segments?
3
a°
A)
68
D)
78
B)
73
C)
76
6
240°
c°
What is the perimeter of an equilateral triangle
inscribed in a circle with circumference 24p ?
b°
A) 36 2
B) 30 3
C) 36 3
D) 24 6
In the figure above, what is the value of a + b + c ?
2
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McGraw-Hill Education: SAT
Exercise Set 1: Geometry (Calculator)
Questions 7–9 are based on the figure below.
11
y
A(3, 15)
y°
B(10, 12)
x
3m
y°
2y°
C(10, 2)
O
2m + 5
x
Note: Figure not drawn to scale.
In the figure above, what is the value of x ?
Note: Figure not drawn to scale.
A) 5 2
7
In the figure above, what is the perimeter of
quadrilateral ABCO, to the nearest integer?
B) 5 3
D) 15 3
C) 15 2
Questions 12–15 are based on the information
below.
In the xy-plane, ABCD is a square. Point A has
coordinates (-1, 2) and point B has coordinates
(3, 5).
8
In the figure above, what is the area, in square
units, of ABCO ?
12
Which of the following could be the coordinates
of C ?
A) (0, 9)
9
C) (2, -2)
B) (6, 0)
D) (-4, 6)
13
In the figure above, point K (not shown) is the mid­point of OA, and point M (not shown) is the
midpoint of AB. What is the slope of KM ?
What is the area of square ABCD ?
A) 25
B) 28
C) 30
D) 32
14
What is the slope of BC ?
A) - 4
B) - 3
C) 3
3
4
4
10
In the xy-plane, point H has coordinates (2, 1) and
point J has coordinates (11, 13). If HK is parallel to
the x-axis and JK is parallel to the y-axis, what it
the perimeter of triangle HJK ?
D) 4
3
15
What is the distance between C and the midpoint
of AB ?
A)
50
4
D)
125
2
B)
125
4
C)
50
2
3
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Distance and Area
1
5
A cereal company sells oatmeal in two sizes of
cylindrical containers. The radius of the larger
container is twice that of the smaller, and the
height of the larger container is 50% greater than
the smaller. If the smaller container holds 10
ounces of oatmeal, how many ounces can the
larger container hold?
Which of the following equations represents a circle
in the xy-plane that intersects the x-axis at (3, 0) and
(9, 0) ?
A) (x - 6)2 + (y - 4)2 = 25
B) (x - 3)2 + (y - 9)2 = 25
C) (x - 6)2 + (y - 4)2 = 36
D) (x - 3)2 + (y - 9)2 = 36
6
2
C
D
P
N
In the figure above, P and N are the centers of the
circles and PN = 6. What is the area of the shaded
region?
Note: Figure not drawn to scale.
In the figure above, CD is tangent to both circles,
which are tangent to each other. If the smaller circle
has a circumference of 4p and the larger circle has
a circumference of 16p, what is the length of CD ?
A) 18π − 9 3
B) 24π − 9 3
C) 24π − 18 3
D) 36π − 18 3
7
y
P (9, k)
3
What is the area, in square inches, of a circle with
diameter 6p2 inches?
A) 9p4
B) 9p5
C) 36p4
D) 36p5
O
4
12
D)
58
B)
48
C)
(9, 0)
x
The diagram above shows a hexagon with all sides
congruent and all angles congruent. What is the
value of k ?
What is the length of the longest line segment that
connects two vertices of a rectangular box that is
6 units wide, 4 units long, and 2 units tall?
A)
(3, 0)
A) 6 2
56
B) 6 3
C) 12 2
D) 12 3
4
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CHAPTER 10 / THE SAT MATH: ADDITIONAL TOPICS
431
Exercise Set 2: Geometry (Calculator)
8
11
The Great Pyramid in Giza, Egypt, has a height
of 140 meters and a volume of 2.6 million cubic
meters. If a scale model of the Great Pyramid is
to be built that is 2 meters high, what will be the
volume, in cubic meters, of this model?
y
D (14, 14)
A(−2, 12)
C (14, 6)
O
x
B(4, 0)
12
Which of the following equations defines a circle
that is tangent to the y-axis?
What is the area, in square units, of the
quadrilateral above?
A) (x - 2)2 + (y + 3)2 = 2
B) (x - 2)2 + (y + 3)2 = 3
C) (x - 2)2 + (y + 3)2 = 4
D) (x - 2)2 + (y + 3)2 = 9
9
What is the degree measure, to the nearest whole
degree, of an angle that measures 5.6 radians?
Questions 13 and 14 refer to the diagram below.
E
F
9
B
10
C
15
A
O
D
G
10
x°
N
The figure above shows two rectangles that share
a common vertex, and AF is a line segment that
passes through C.
M
13
(
In the figure above, arc MN has a length of 11.5.
To the nearest integer, what is the value of x ?
What is the ratio of the area of rectangle ABCD to
the area of rectangle AEFG ?
A) 3:5
B) 9:25
C) 5:8
D) 25:64
5
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McGraw-Hill Education: SAT
14
15
If CD = 9, what is the perimeter of rectangle AEFG ?
A) 67.2
x°
B) 72.6
C) 76.2
4x°
O
D) 78.6
8
Point O is the center of the circle above. What is the
area of the shaded region?
A) 64p -16
3
16
p
B)
-8
3
C) 64p -12
3
64
p
D)
-8
3
3
3
3
3
6
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Understanding Basic Trigonometry
1
6
What is the greatest possible value of f if
8 sin 2 x 1
f (x) =
− ?
2
2
If sin x =
expressions is equal to b ?
a
A)
2
B)
2
If cos
p
a
and 0 < x < , which of the following
b
2
π
a
= a , what is the value of
?
⎝3⎠
⎝3 ⎠
sin
1
⎝x⎠
1
π
cos − x
⎝2
⎠
C) 1 - sin2 x
D)
sin
π
−x
⎝2
⎠
3
7
If (sin x - cos x)2 = 0.83, what is the value of
(sin x + cos x)2 ?
If sin b = a, which of the following could be the
value of cos (b + p) ?
a2 - 1
A)
B) a2 - 1
4
π
⎝6 ⎠ ?
Which of the following is equivalent to
π
cos
⎝ 3⎠
C)
sin
1
B)
6
A)
1
C)
3
3
2
- 1 - a2
D) 1 - a2
D) 1
8
If 0 < x <
5
p
cos x
3
and
= , what is the value of
2
2
1 − sin x 2
cos x ?
y
II
I
x
O
III
A)
1
9
B)
1
3
C)
4
9
D)
2
3
IV
If sin q < 0 and sin q cos q < 0, then q must be in
which quadrant of the figure above?
A)
I
B)
II
C)
III
D)
IV
7
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Understanding Complex Numbers
1
6
If a + bi = (1 + 2i)(3 - 4i), where a and b are
constants and i = −1, what is the value of a + b ?
Which of the following expressions is equal to
(2 + 2i)2 ?
A) 0
B) 4i
C) 8i
D) 4 - 4i
2
If a + bi =
4+i
, where a and b are constants and
2−i
7
If B(3 + i) = 3 - i, what is the value of B ?
i = −1, what is the value of a ?
A)
B)
C)
3
For what value of b does (b + i)2 = 80 + 18i ?
D)
3 4
+ i
5 5
4 3
+ i
5 5
3 4
- i
5 5
4 3
- i
5 5
8
x2 + kx = -6
4
The solutions of the equation x2 - 2x + 15 = 0 are
If one of the solutions to the equation above is
x = 1 − i 5, what is the value of k ?
x = a + i b and x = a − i b , where a and b are
A) -4
positive numbers. What is the value of a + b ?
B) -2
C) 2
D) 4
9
5
If im = -i, which of the following CANNOT be the
value of m ?
Given that i = −1, which of the following is equal
to
1
?
(1 + i )2
A)
1 1
- i
2 2
1
- i
2
1
i
2
1 1
+ i
2 2
B)
C)
D)
A) 15
B) 18
C) 19
D) 27
8
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