What is the area of rectangular region R ?
(1) Each diagonal of R has length 5.
(2) The perimeter of R is 14.
What's the best way to determine whether statement 1 is sufficient? Can any experts explain?
What is the area of rectangular region R ?
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Let L = length, W = width, D = diagonal, P = perimeter, A = area.ardz24 wrote:What is the area of rectangular region R ?
(1) Each diagonal of R has length 5.
(2) The perimeter of R is 14.
In any rectangle:
LÂ² + WÂ² = DÂ².
2L + 2W = P.
LW = A.
Statement 1:
Thus:
LÂ² + WÂ² = 5Â².
Case 1: L=3 and W=4, with the result that 3Â² + 4Â² = 5Â²
In this case, A = LW = 3*4 = 12.
Case 2: L=âˆš24 and W=1, with the result that (âˆš24)Â² + 1Â² = 5Â²
In this case, A = LW = (âˆš24)(1) = âˆš24.
Since the area can be different values, INSUFFICIENT.
Statement 2:
Thus:
2L + 2W = 14
L+W = 7.
Case 1: L=3 and W=4, with the result that 3+4 = 7
In this case, A = LW = 3*4 = 12.
Case 3: L=6 and W=1, with the result that 6+1=7
In this case, A = LW = 6*1 = 6.
Since the area can be different values, INSUFFICIENT.
Statements combined:
Only Case 1 satisfies both statements.
In Case 1, A=12.
SUFFICIENT.
The correct answer is C.
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Solution:BTGmoderatorAT wrote: ↑Mon Jan 22, 2018 7:24 pmWhat is the area of rectangular region R ?
(1) Each diagonal of R has length 5.
(2) The perimeter of R is 14.
What's the best way to determine whether statement 1 is sufficient? Can any experts explain?
Question Stem Analysis:
We need to determine the area of region R, which is a rectangle.
Statement One Alone:
Knowing the length of the diagonal of a rectangle is not sufficient to determine its area. For example, if R is a square (recall that a square is also a rectangle), it’s side length would be 5/√2 and its area would be (5/√2)^2 = 25/4. However, R could also be a nonsquare rectangle with dimensions 3 and 4. In this case, the area of R is 3 x 4 = 12. Statement one alone is not sufficient.
Statement Two Alone:
Knowing the perimeter of a rectangle is not sufficient to determine its area. For example, if R has dimensions 2 and 5, its area is 2 x 5 = 10. However, if R has dimensions 3 and 4, its area is 3 x 4 = 12. Statement two alone is not sufficient.
Statements One and Two Together:
If we let L and W be the dimensions of the rectangle, we can create the equations:
L^2 + W^2 = 5^2
and
2L + 2W = 14
We see that we can solve L and W based on the equations we’ve set up, and once we have the values of L and W, then the area of region R is just the product of L and W. Both statements together are sufficient. (Note: L = 3 and W = 4 OR L = 4 and W = 3. Either way, the area of R is 12.)
Answer: C
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