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

This is a (C) trap question designed in a way such that a test-taker may be tempted to believe that both statements are necessary for answering the question when, actually, only one is necessary.

Statement (1)

This statement tells us two things: that the figure is a rectangle and that the area of the rectangle is 54.

We know that the area of a rectangle = b x h. So, b x h = 54.

Now, we have to find the area of the shaded region, and here's how the GMAT tempts us to choose the wrong answer: it breaks the shaded region into two triangles to make us start to wonder whether there are multiple possible areas of the shaded region, with the sizes of the different areas depending on how we draw those two triangles.

So, to correctly answer a question like this one, we have to be sure to see that the shapes of the two shaded triangles don't matter.

Why don't they matter? For one thing, regardless of the shape of the two shaded triangles, the large, unshaded triangle has the same base and height as the rectangle. Since the area of a triangle is always (b x h)/2, the area of the large triangle will always be half the area of the rectangle, regardless of the shapes of the two smaller triangles.

So, regardless of the shapes of the two smaller triangles, the area of the shaded region also will always be half of the area of the rectangle.

So, Statement (1) is sufficient.

Statement (2)

Of course, we can easily tell that Statement (2) is insufficient, as it lacks any information on the area of anything in the figure, but a tricky aspect of Statement (2) is that it is designed to support the narrative that we need to have information on 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.

So, Statement (2) is designed to line up with the faulty thinking that the size of the area of the shaded region depends on the shapes of the two shaded triangles and tempt us to think that we need both statements and choose choice (C), as many people who have answered this question have.

So, the correct answer to this question is (A), and arriving at that answer requires avoiding being fooled by the way the question is constructed and seeing that, no matter where point E is on the base of the rectangle, the area of the shaded region is the same.
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Let AD=a , AE = x And ED = y And the height =h

1) as ABCD rectangle , areaABCD = a.h = (x+y).h= 54
area of the shaded region is Area of triangle ABE + Area of triangle CDE = \(\frac{1}{2} x h\)+ \(\frac{1}{2} yh\) = \(\frac{1}{2} (x+y) h\) =\(\frac{1}{2} *54\)=27 ..........................(S)

2) Not sufficient , as it is not mentioned what kind of quadrilateral. The area properties will differ in (square, rhoumbus, parallellogram , rectangle ) vs trapepezoids, irregular quadrilateral)......(NS)

Hence the answer is A
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OA : A
Statement 1 : The area(\(b*h\)) of the rectangle ABCD is 54.
The area(\(\frac{1}{2}*b*h\)) of the unshaded triangle= \(\frac{1}{2}* 54=27\)
Area of shaded region = rectangle area- unshaded triangle area\(= 54-27 = 27\)
Sufficient
Statement 2 : Shape of ABCD is not defined , area of shaded area will vary according to shape of ABCD
Not sufficient

Target question: What is the area of the shaded region

Statement 1: The area of the rectangle ABCD is 54
Let's label the two shaded triangles as ∆1 and ∆2.
Also, let j be the length of the base of ∆1 and k be the length of the base of ∆2.
Let h = the height of both triangles (since ABCD is a RECTANGLE, the height is consistent for both ∆s)


Area of ∆ = (1/2)(base)(height)
So, (area of ∆1) + (area of ∆2) = [(1/2)(j)(h)] + [(1/2)(k)(h)]
= (1/2)(h)[j + k] [I factored out the (1/2)(h)]

IMPORTANT: j + k = the length of the BASE of rectangle ABCD.
In other words, (area of ∆1) + (area of ∆2) = (1/2)(h)[BASE of rectangle ABCD]
Since (h)(the BASE of rectangle ABCD) = the area rectangle ABCD, we can say that.....
(area of ∆1) + (area of ∆2) = HALF the area of rectangle ABCD

Since statement 1 tells us that the area of the rectangle ABCD is 54, we can conclude that (area of ∆1) + (area of ∆2) = (1/2)(54) = 27
Since we can answer the target question with certainty, statement 1 is SUFFICIENT


Statement 2: AE = 2ED
IMPORTANT: For geometry Data Sufficiency questions, we are typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the video below

This technique can save a lot of time.

Here, we are told that line segment AE is TWICE the length of line segment ED.
Is this enough information to LOCK IN the combined areas of the two shaded triangles? NO.
Notice that statement 2 does not prohibit us from making rectangle ABCD are TALL or as SHORT as we want.


By making rectangle ABCD are TALL or as SHORT as we want, we can make the shaded area as large or as small as we wish.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

RELATED VIDEO

Easy 30 second question if you remember that by drawing a triangle within a rectangle or square, where:
1) the base is the entire line segment of the figure
2) the height touches the opposite line segment

You divide the area of that figure by half

Statement 1:
When a triangle inscribed in a rectangle shares a side with the rectangle, the area of the triangle is equal to half the area of the rectangle.
Here, inscribed triangle BCE and rectangle ABCD share side BC.
Thus:
\(BCE = \frac{1}{2}ABCD = \frac{1}{2}*54 = 27\), with the result that the shaded region = rectangle ABCD - triangle BCE = 54-27 = 27.
SUFFICIENT.

Statement 2:
Since ED can be any nonnegative value, the area of the shaded region cannot be determined.
INSUFFICIENT.


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The same concept was tested on the SweSat in 2016.

Here is a copy of that question.

ABCD is a rectangle. E is a point on the side BC. If BC is 4 cm, what is the area of the triangle AED?

(1) CD is 19/8 of AD

(2) BE = EC
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Dear Sir,

I had a general DS question. May we assume that the triangle (in this question) touches the side of the rectangle? Could there be a possibility that the tip of the triangle could leave the tiniest possible gap?

Hi BrentGMATPrepNow, can we use "Lock" concept for St.1 also? Which mean ABCD area vlaue lock in the shaded area value? Thanks

Statement A is sufficient
1/2*AE*h + 1/2*ED*h
1/2*h*[AE+ED]
1/2*h*b= 1/2*54

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