M13-05

Question

What is the area of parallelogram  ABCD ?

(1)  AB = BC = CD = DA = 1 
 
(2)  AC = BD = \sqrt{2}

Bunuel · Answer

Official Solution:

What is the area of parallelogram  ABCD ? 
 
Notice that we are told that  ABCD  is a parallelogram.
 
(1)  AB = BC =CD = DA = 1 . 
 
Since all four sides of the parallelogram  ABCD  are equal, we can deduce that  ABCD  is a rhombus. The area of a rhombus is given by the formula  \frac{d_1*d_2}{2}  (where  d_1  and  d_2  are the lengths of the diagonals) or alternatively  bh  (where  b  is the length of the base and  h  is the height). However, we do not have sufficient data to calculate the area in this case. Not sufficient.  It's important to note that, while all squares are rhombi, the converse is not necessarily true. Therefore, from this statement, we can deduce that  ABCD  could be a square, but it's not necessarily a square. 
 
(2)  AC = BD = \sqrt{2} . 
 
Given that the diagonals of the parallelogram  ABCD  are equal, we infer that  ABCD  is a rectangle. The area of a rectangle is calculated as  length * width . Once again, we don't have the necessary data to determine the area. Not sufficient.  It's important to note that the area of a rectangle cannot be calculated solely from the length of its diagonal. 
 
(1)+(2) Considering both conditions,  ABCD  is both a rectangle and a rhombus, which implies that it is a square. Hence, the area can be calculated as  	ext{area} = 	ext{side}^2 = 1^2 = 1 . Sufficient.

Answer: C

bigzoo · Answer

stmnt 2: could you help me please why the (d1*d2)/2 formula cannot be used for calculating the are of a rectangle?

Bunuel · Answer

First of all, two diagonals of a rectangle are equal in length so if this formula could be applied to rectangles, then it would be area=d^2/2. But this is NOT correct. Consider this: changing angles between diagonals would change its area, so a rectangle with the diagonal of d can have infinitely many different areas, depending on the angle between diagonals.

Hope it's clear.

P.S. With d^2/2 you can find the area of only one specific rectangle, namely a square: square area = d^2/2.

gmatkiller88 · Answer

Bunuel,
So far I only knew that area of a square = (side)^2 but as per the information you have given above can we find the area if we are given the length of diagonal without really finding out the length of side in a square?

What I mean is suppose we are given length of a diagonal in a square as root 2, then do we need to find the length of sides (by using property of isoceles right triangle) or can we just do d^2/2 and find the area. I can see that both ways the area is same in this example however can you please confirm that this always works? Many thanks.

Bunuel · Answer

Yes, you can ALWAYS find the area of a square by diagonal^2/2.

toby001 · Answer

This might be a really stupid question; but if a parralelogram has all 4 sides equal, doesn't that make it a square?

Thanks

Bunuel · Answer

Not necessarily, it might also be a rhombus. Check for more here: math-polygons-87336.html

tinyinthedesert · Answer

No formulas, calculations or even notes are required here - it can be solved visibly in about 30 seconds with basic geometry rules
 
Remember that a square is also a rectangle, a parallelogram and a rhombus, and a rectangle is also a parallelogram. To maximise the area for any given perimeter, form a square

(1) tells us the perimeter, and that the sides are equal. We can make various rhombus shapes from this information, from a square (max area) to a very narrow diamond (smaller area). Minimum area is zero, when we squash the diamond so flat it becomes a line. In any case, there are many possible areas so we cannot determine. Insufficient
(II) This information tells us the diagonals are equal. This means that the parallelogram is in fact a square or a rectangle (geometry rules). Insufficient

(i) and (ii) - we have must have square with sides equal to 1 and diagonals equal to rt2. Area is 1. Sufficient

Basically 1 tells us we that the parallelogram that is a rhombus, and 2 tells us that that the parallelogram that is a rectangle. Combining them, we must have a square, as only a square is both a rectangle and a rhombus. We actually only need the diagonal or the side value to calculate the area in this case

Remember the rules - 
Square - sides and diagonals equal
Rectangle - diagonals equal
Rhombus - sides equal
All are parallelograms

TBT · Answer

Hi,
My answer was correct but my reasoning for 1 not sufficient was : That it can rhombus or square. Is this sufficient to eliminate option 1?

Bunuel · Answer

Well, if ABCD is a square then the area would be 1 but if ABCD is a rhombus (but not a square), the area would be different from 1.

Bunuel · Answer

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

shauryavats02 · Answer

---------------
 Hi  Bunuel  , the diagonals of a rhombus and square bisect each other at 90 degrees. Therefore isn't statement 1 sufficient? The 4 triangles formed by the diagonals of the rhombus have the hypotenuse (the side) as 1 m and since rest of the angles are 45 & 45 degrees (since the diagonals are equal and bisect each other), we can use the 45-45-90 ratio to finally find the diagonals & hence the area.

Bunuel · Answer

It's accurate that the diagonals of both a rhombus and a square bisect each other at 90 degrees. However, asserting that the "rest of the angles are 45 & 45 degrees" is only correct for a square. For a generic rhombus that isn't a square, this assumption doesn't hold. You can try drawing both to see the difference.

M13-05

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