Sat math Essentials

3.
e.
Perpendicular lines have slopes that are negative
reciprocals of each other. The slope of the line
given is 
2
3
. The negative reciprocal of
2
3
is –
3
2
.
Every line with a slope of –
3
2
is perpendicular to
the given line;
y
= –
3
2
x
+ 5 is perpendicular to 
y
= 
2
3
x
– 5.
4.
b.
If
r
= 30, 30% of
r
= (0.30)(3) = 9. 9 is equal to
75% of
s
. If 0.75
s
= 9, then 
s
= 12. 50% of
s
=
(0.50)(12) = 6.
5.
b.
30 men 
42 square feet = 1,260 square feet of
space; 1,260 square feet ÷ 35 men = 36 square
feet; 42 – 36 = 6, so each man will have 6 less
square feet of space.
6.
d.
The order of the four songs is important. The
orderings A, B, C, D and A, C, B, D contain the
same four songs, but in different orders. Both
orderings must be counted. The number of six-
choose-four orderings is equal to (6)(5)(4)(3)
= 360.
7.
a.
The statement “Raphael runs every Sunday” is
equivalent to “If it is Sunday, Raphael runs.”
The contrapositive of a true statement is also
true. The contrapositive of “If it is Sunday,
Raphael runs” is “If Raphael does not run, it is
not Sunday.”
8.
c.
Line 
AB
is perpendicular to line 
BC
, which
makes triangle 
ABC
a right triangle. Angles 
DAF
and 
DCH
are alternating angles—angles made
by a pair of parallel lines cut by a transversal.
Angle 
DAF
angle 
DCH
, therefore, angle 
DCH
= 120 degrees. Angles 
DCH
and 
ACB
form a
line. There are 180 degrees in a line, so the meas-
ure of angle 
ACB
= 180 – 120 = 60 degrees. Tri-
angle 
ABC
is a 30-60-90 right triangle, which
means that the length of the hypotenuse,
AC
, is
equal to twice the length of the leg opposite the
30-degree angle,
BC
. Therefore, the length of
BC
is 
1
2
0
, or 5. The length of the leg opposite the 60-
degree angle,
AB
, is 
3
times the length of the
other leg,
BC
. Therefore, the length of
AB
is
5
3
.