M03-02

Official Solution:


Is quadrilateral \(J\) a square?

(1) All sides of \(J\) are equal.

Quadrilateral \(J\) could be a square, but it could also be a rhombus since both shapes have equal sides. This information alone is not sufficient.

(2) The area of \(J\) is a multiple of 10.

Knowing that the area is a multiple of 10 does not provide enough information about the shape of the polygon. This information is not sufficient.

(1)+(2) We could have a square with an area that is a multiple of 10, but we could also have a rhombus with an area that is a multiple of 10. The information is still not sufficient to determine if \(J\) is a square.


Answer: E
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I think this is a good quality question. It seems easy but you need to pay attention on details.

Hi,

Maybe a dumb question, but can a shape be both a rhombus and a square?

Yes. All squares are rhombi but not all rhombi are squares.
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I chose C, because - I was not able to find any area of square matching with area of rhombus, area which should be multiple of 10.

Area of Square = \(x^2\)
Area of Rhombus =\( \frac{Product of Diagonals}{2}\)

I am not able to find the equation for area of Rhombus in terms of side? Can someone please help here??

VeritasKarishma @Banuel CrackVerbal
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WarriorWithin

Note that area of a quadrilateral cannot tell you anything about the kind of quadrilateral it is until and unless you are given that the sides must be integers. Even then, you get very limited info.

If area of a quadrilateral is 100, it could be a square, rectangle, rhombus, parallelogram, trapezoid or no special quadrilateral.
It doesn't matter whether you are able to find area in terms of length of sides.

Say a rhombus with area 100 could have diagonals 50 and 40 so that area = 50*40/2 = 100.
Then the side would be \(\sqrt{25^2 + 20^2}\) i.e. 32.0156
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I think this is a high-quality question and I agree with explanation.

Sorry KarishmaB , I think there's a typo: 50*40/2 is 1000, right?

Hi WarriorWithin
Let me rewrite the first aspect of your question here
"I am not able to find the equation for area of Rhombus in terms of side?"
Note that a square has the properties of a rhombus and in addition it also has equal angles.
Diagonals of a square of side, 'a' are equal and each equal to \(\sqrt{ a^2+ a^2 }\) = \(\sqrt{ 2 }a\)
Hence the area of a square (which is also a special rhombus)is derived from this
1/2 * d1 * d2
= 1/2 * \(\sqrt{ 2 }a\) * \(\sqrt{ 2 }a\)
= \(a ^ 2\)

Lets look at the second aspect now-
While you chose option C, you have assumed that the quadrilateral is a square and has area as a multiple of 10. Although we can have a square of side 20 units with area 400 square units, we can also have other quadrilaterals that can have area as a multiple of 10 and are not a square. Say a rhombus(all sides equal) with area 200 and with diagonals as 10 and 40 units so that the product of the diagonals = 10 * 40 =400 and receive an answer of No for the quadrilateral to be a square.
Hope this clears the query.

Devmitra Sen
GMAT Mentor

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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