What is the area of rectangular region R ?

[BTGmoderatorAT]

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?

[GMATGuruNY]

What is the area of rectangular region R ?

(1) Each diagonal of R has length 5.
(2) The perimeter of R is 14.

Let L = length, W = width, D = diagonal, P = perimeter, A = area.
In any rectangle:
L² + W² = D².
2L + 2W = P.
LW = A.

Statement 1:
Thus:
L² + W² = 5².
Case 1: L=3 and W=4, with the result that 3² + 4² = 5²
In this case, A = LW = 3*4 = 12.
Case 2: L=√24 and W=1, with the result that (√24)² + 1² = 5²
In this case, A = LW = (√24)(1) = √24.
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.

[Scott@TargetTestPrep]

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?

Solution:

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 non-square 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