Question: What is the area of the shaded region in the figure shown?

(1) The area of the rectangle ABCD is 54.
(2) AE = 2ED

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Answer: A

Solution and Explanation:

Approach Solution 1:

Statement (1)

This assertion reveals two facts about the figure: first, that it is a rectangle, and second, that the area of the rectangle is 54.

We are aware that b x h equals the area of a rectangle. So, b x h = 54.

Now that we have to determine the size of the shaded region, here is how the GMAT tempts us to select the incorrect response: it divides the shaded region into two triangles to make us begin to wonder if the shaded region has more than one possible area, with the sizes of the various areas depending on how we draw those two triangles.

Therefore, we must be certain to understand that the shapes of the two shaded triangles are irrelevant in order to appropriately respond to a question like this one.

Why are they irrelevant? For one thing, the huge, unshaded triangle has the same base and height as the rectangle regardless of the design of the two shaded triangles. No matter how the two smaller triangles are shaped, the area of the huge triangle will always be equal to half the area of the rectangle since the area of a triangle is always (b x h)/2.

Therefore, the area of the darkened zone will always be equal to half of the area of the rectangle, regardless of the shapes of the two smaller triangles.

Therefore, Statement 1 alone is adequate.

Statement (2)

A tricky aspect of Statement (2) is that it is intended to support the claim that we need knowledge of the shapes of the two smaller triangles in order to determine the area of the shaded region, as it provides some information on those shapes. Of course, we can easily tell that Statement (2) is insufficient because it contains no information on the area of anything in the figure.

Therefore, Statement (2) is intended to support the fallacious belief that the size of the shaded zone depends on the shapes of the two shaded triangles. This is done to lure us to believe that we require both statements and select Option (C), as many respondents to this question have done.

The proper response to this question is, therefore, (A), but getting there requires not being deceived by the way the question is written and realizing that the area of the shaded zone is constant regardless of where point E is located on the base of the rectangle.

Approach Solution 2:

Statement 1: The rectangle ABCD's surface area is 54

Let's assign the numbers 1 and 2 to the two shaded triangles.

Let j and k be the lengths of the bases of 1 and 2, respectively.

Since ABCD is a RECTANGLE, the height is constant for both s. Let h = the height of both triangles.

Area of triangle = (1/2)(base) (height)

In other words, (area of 1) + (area of 2) = [(1/2)(j)(h)] + [(1/2)(k)(h)]

= (1/2)(h)[j + k] [I eliminated the (1/2)(h)]

Important: The length of the BASE of the rectangle ABCD is equal to j + k.

Alternatively, (area of 1) + (area of 2) = (1/2). (h)

[BASE of ABCD rectangle]

We can state that (area of 1) + (area of 2) = HALF the area of rectangle ABCD because of (h)(the BASE of rectangle ABCD) = the area rectangle ABCD.

We can deduce that (area of 1) + (area of 2) = (1/2)(54) = 27 because statement 1 states that the area of the rectangle ABCD is 54.

Statement 1 is sufficient because we are confident in our ability to respond to the target question.

Second premise: AE = 2ED

Important: When examining a geometry question's Data Sufficiency, we usually look to see if the assertions "lock" a certain angle, length, or form into having only one potential measurement.

A is the correct answer.

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