Quadrilaterals
B
C
A
D
Note: Figure not drawn to scale.
1. In figure above, ABCD is a parallelogram, BD = 8 and AD = 10. What is the area of ABCD?
(A) 24
(B) 48
(C) 60
(D) 72
(E) Cannot be determined from the information given.
R
S
T
U
Note: Figure not drawn to scale.
2. In the figure above, RSTU is a square. If RU = 10, what is the perimeter of RSTU?
(A) √
(B) 20
(C) √
(D) 50
(E) 100
3. In a quadrilateral, if two of the angles measure
other two angles?
and
, what could be the measure of the
(A)
(B)
(C)
(D)
(E) It cannot be determined from the information given.
7
3
6
4.5
4.
What is the area of the figure above?
(A) 27
(B) 34.5
(C) 36.5
(D) 42
(E) It cannot be determined from the information given.
A
B
C
D
5. If a circle is inscribed in square ABCD above and only touches the square at four point, what is
the area of the shaded region if AB = 4?
(A)
(B)
(C)
(D)
(E)
Answers and Explanations
1. The correct answer is B. The most efficient way to solve this problem is to determine the area of
the two equal triangles contained within the parallelogram. Adding their area together will give
you the area of the parallelogram. The triangles are right triangles and, since we know the length
of two of the sides (one leg and the hypotenuse), we can use the Pythagorean Theorem to solve
for the length of the third side. First, let’s make life easier and redraw (and reorient) one of the
triangles. This is just a rough drawing, but it doesn’t have to be perfect!
D
8
B
10
A
Using the Pythagorean Theorem, we can determine AB:
From here, we have the base and height of the triangle. Thus, the area is
. Notice,
this is answer choice A. Sneaky! But we want the area of the parallelogram, so we must double it.
Thus, the answer is 48, choice B.
2. The correct answer is C. If the diagonal of square RSTU is 10, we can determine the sides of the
square by recognizing that the diagonal creates two 45-45-90 degree triangles. If RU = 10, all the
sides must equal √ . We know this because of the Pythagorean Theorem (
).
Thus, we can calculate the perimeter by adding up the length of all 4 sides. Since all 4 sides are
equal, the perimeter is √
, or √ . Be careful and remember what the question is asking
for. If you calculate the area instead of the perimeter, you would get 50, which is not so
conveniently choice D.
3. The correct answer is D. Don’t be tempted by choice E! While it is true that we have no way of
knowing what the actual values of the two angles are, the question simply asks for two possible
values. For the values to be possible, they must, together with the two angle measurements we are
given, add up to 360 degrees. Since
, we have 252 degrees unaccounted for. Thus,
for an answer choice to be correct, the values given must add up to 252. Choice D is the only one
that works (
).
4. The correct answer is B. Again, as tempting as it may be to opt for choice E, you do have
enough information given to solve the problem. When faced with a shape that is irregular, divide
it into multiple shapes. We’ve split it into two rectangles by adding a single line below. We can
add in the measurements of the missing segments and then find the area of each rectangle.
7
3
2.5
6
3
4.5
The area of the small rectangle on the left is
or 7. The area of the larger rectangle on the
right is
or 27. The area of the whole figure is simply the sum of the two:
.
5. The correct answer is E. The area of the shaded region is just the area of the square minus the
area of the circle. The area of the square is simply or 16, since s = 4. To find the area of the
circle, we need to know it’s radius. Since we are only given a measurement for the square, we
need to figure out what measurement they have in common. In this case, the diameter of the circle
is equal to the side of the square (4). The radius of a circle is half the diameter. Thus, the radius
is 2. We can plug this into the formula for the area of a circle (
) to get . The area of the
shaded region is simply
, choice E.