Explanations
1. C
Because lines l and m are
parallel and line AB is a transversal, the angle
whose measure is labeled as 120º is supplementary to 
.
Now we have a 30-60-90 triangle whose longer leg, AC,
is also the distance between lines l and m.
Using the 
:
:
side ratios for 30-60-90 triangles
you can use the hypotenuse length to calculate the lengths of the
other two legs. The short leg has a 
:
ratio to the hypotenuse, so its
length is 5/2.
The long leg has a 
:
ration to the short leg, so its
length is 5
⁄2.
2. E
Let’s analyze each statement separately.
-
This statement implies that DF = EF.
We know that BF = AF because it
is given that line CF is the perpendicular bisector
of AB, and by definition, F is the
midpoint of AB. It is also given that the area
of triangle CDB is equal to the area of triangle CEA.
These two triangles share the same height, and since the area of
a triangle is found by the formula 1⁄2 b
h, it follows that
if their areas are equal, their bases are equal too. If BD
= AE, then by subtracting DE from each
segment, we have BE = AD and thus EF =
DF. So statement I is true.
- This
statement is simply not backed by any evidence. All we know is that BE = AD, EF = DF,
and BF = AF. As long as points E and D are
equidistant from F, all these conditions hold, so there is no guarantee
that they are the midpoints of BF and AF,
respectively. E and D could be
anywhere along BF and AF, respectively,
as long as they are equidistant from F. Thus, this statement
is not necessarily true.
- Triangles CDB and CEA are
equal in area; this is given. By subtracting the area of triangle CED from
each of these triangles, we see that triangles CEB and CDA must
have the same area. This statement is true.
Only statements I and III must be true.
3. C
The area of a triangle with base x and
height h is given by the formula 1⁄2xh.
The area of a square with sides of length x is x2.
Since you know the two shapes have equal areas, you can set the
two expressions equal to each other and solve for h:
The correct answer is h = 2x.
4. D
If ABD is an equilateral triangle,
then AD = AB = BD =
4, and all the sides of the rhombus have a length of 4 (by definition
of a rhombus, all sides are congruent). Also, by definition of a
rhombus, opposite angles are congruent, so 
. Draw an altitude from a to DC to
create a 30-60-90 triangle, and from the length ratio of x : 
x : 2x among
the sides, you can calculate the length of this altitude to be 2
. The area of a rhombus is bh,
so the area of this rhombus is 4 
2
= 8
.
5. D
The length of the arc depends on the circumference
of the circle and the measure of the central angle that intercepts
that arc. The formula is:
where n is the measure of the central
angle that intercepts the arc and r is the radius.
Angle c is the inscribed angle or
one-half as large as the central angle that intercepts the circle
at the same points. So the measure of this angle is 2cº.
Now simply plug the values into the formula: the length
of arc AB is:
2π(8)
= 
= 
