Correct Answer: C
Explanation:
Geometry Rectangles; Perimeter; Area
The perimeter of a rectangle is equal to 2 times
the rectangle's length plus 2 times the rectangle's
width, or p = 21 + 2w. The diagonals of a rectangle
are equal. In a rectangle, because a diagonal forms
a right triangle, the length of a diagonal is equal
to the square root of the length squared plus the
width squared, or d = √ t2 + w 2.
- (1) If a diagonal = 10, then 10 = √ t2 + w 2, or,
by squaring both sides, 100 = I2 + w2. Without
knowing the value or the relationship between
the other two sides of the right triangle, it is
impossible to solve for / or w, and thus for the
perimeter of the rectangle; NOT sufficient.
- (2) If the area of the rectangle is 48, then it can
be stated that I w = 48. However, without further information, the perimeter cannot be determined.
For example, / could be 6 and w could be 8, and the perimeter would then be 12 + 16 = 28. However, it could also be that / is 4 and w is 12,and in that case the perimeter would be 8+ 24 =
32; NOT sufficient.
Using (1) and (2) together, it is possible to solve
this problem. Since from (2)I w= 48, then
w = 48/I. Substituting this into 100 =I2 +w2 from (1) the equation can be solved as follows:
100 = I2 + (48/I)2 + substitution
100 I2 = I4 + 2,304 multiply both sides by I2
l4 - 100 I2 + 2,304 = 0 move all terms to one side
(l2 - 64)(I2 - 36) =0 factor like a quadratic
I2 =64,I2 = 36 solve for I2
Since I is a length, it must be positive, so I is 48 either 8 or 6. When I = 8, w = -5- = 6, and when
I = 6, w = -48/6 = 8, both of which give the same perimeter.
The correct answer is C; both statements together are sufficient.