Sat math Essentials

9.
b.
Every face of a cube is a square. The diagonal of
a square is equal to 
s
2
, where 
s
is the length of
a side of the square. If
s
2
= 4
2
, then one
side, or edge, of the cube is equal to 4 in. The
volume of a cube is equal to 
e
3
, where 
e
is the
length of an edge of the cube. The volume of the
cube is equal to (4 in)
3
= 64 in
3
.
10.
a.
A line with a 
y
-intercept of –6 passes through the
point (0,–6) and a line with an 
x
-intercept of 9
passes through the point (9,0). The slope of a
line is equal to the change in 
y
values between
two points on the line divided by the change in
the 
x
values of those points. The slope of this line
is equal to 
0
9
–
–
(–
0
6)
= 
6
9
= 
2
3
. The equation of the
line that has a slope of
2
3
and a 
y
-intercept of –6
is 
y
= 
2
3
x
– 6. When 
x
= –6,
y
is equal to 
2
3
(–6) –
6 = –4 – 6 = –10; therefore, the point (–6,–10)
is on the line 
y
= 
2
3
x
– 6.
11.
a.
If
m
< 
n
< 0, then 
m
and 
n
are both negative
numbers, and 
m
is more negative than 
n
. There-
fore, –
m
will be more positive (greater) than
–
n
, so the statement –
m
< –
n
cannot be true.
12.
b.
If
r
is the radius of this circle, then the area of this
circle,
π
r
2
, is equal to four times its circumference,
2
π
r
:
π
r
2
= 4(2
π
r
),
π
r
2
= 8
π
r
,
r
2
= 8
r
,
r
= 8 units. If
the radius of the circle is eight units, then its cir-
cumference is equal to 2
π
(8) = 16
π
units.
13.
a.
Since all students take the bus to school, anyone
who does not take the bus cannot be a student.
If Courtney does not take the bus to school,
then she cannot be a student. However, it is not
–
T H E S AT M AT H S E C T I O N
–
2 3 9

---

necessarily true that everyone who takes the bus
to school is a student, nor is it necessarily true
that everyone who is not a student does not take
the bus. The statement “All students take the
bus to school” does not, for instance, preclude
the statement “Some teachers take the bus to
school” from being true.
14.
a.
Lines 
OF
and 
OE
are radii of circle 
O
and since
a tangent and a radius form a right angle, trian-
gles 
OFH
and 
OGE
are right triangles. If the
length of the diameter of the circle is 24 in, then
the length of the radius is 12 in. The sine of
angle 
OHF
is equal to 
1
2
2
4
, or 
1
2
. The measure of
an angle with a sine of
1
2
is 30 degrees. Therefore,
angle 
OHF
measures 30 degrees. Since angles
BGH
and 
OHF
are alternating angles, they are
equal in measure. Therefore, angle 
BGH
also
measures 30 degrees.