What is the area of parallelogram ABCD ?

Yes, both a rhombus and a square have equal sides. From (1) we know that ABCD is a rhombus. A square is a special type of a rhombus, so from (1) ABCD is a rhombus and can be a square.

What is the area of parallelogram \(ABCD\)?

Notice that we are told that ABCD is a parallelogram.

(1) \(AB = BC =CD = DA = 1\) --> all four sides of parallelogram ABCD are equal, which implies that ABCD is a rhombus. Area of a rhombus equals to \(\frac{d_1*d_2}{2}\) (where \(d_1\) and \(d_2\) are the lengths of the diagonals) or \(bh\) (where \(b\) is the length of the base and \(h\) is the altitude), so we don't have enough data to calculate the area. Not sufficient.

(2) \(AC = BD = \sqrt{2}\) --> the diagonals of parallelogram ABCD are equal, which implies that ABCD is a rectangle. Area of a rectangle equals to length*width, so again we don't have enough data to calculate the area. Not sufficient. Notice that you cannot find the area of a rectangle just knowing the length of its diagonal.

(1)+(2) ABCD is a rectangle and a rhombus, so it's a square --> area=side^2=1^2=1. Sufficient.

Answer: C.

Hope it's clear.
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1. Can be square or rhombus.

2.
Diagonals are same for rectangle and square.

For square the area will be:
Area = 1*1 as the side will be 1. Diagonal is \(sqrt{2}\), Diagonal=hypotenuse of 45-90-45 right triangle. Side= 1.

For rectangle the sides can be:
0.5, 1.12; Area = 0.56
OR
0.75, 1.2; Area = 0.9

Basically, all combination of l and w that satisfies:
l^2+w^2=2. And there are infinite such possibilities.

Combining;
We know it's a square and area is 1.

Ans: "C"

Bunuel,
A rhombus and a square with same lengths have different areas?
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Good question.

If you squeeze a square along its diagonal you'll get a rhombus. Different rhombuses you'll get while doing that, will have different area. So, the answer to your question is yes.

Hope it's clear.
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Pardon me. I didn't get you, Bunuel..

Are you saying that the square and different shapes of rhombuses with same length will have different areas?
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Yes, that's what I'm saying.
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ok thanks..
Now 2 questions:
1)if the square and different shapes of rhombuses with same length will have different areas, the square will have the largest area . Guess this is correct.

2)this question seems dubious now to me.. A square is also a parallelogram and even a rhombus is.. so how can we be sure that ABCD is not a square and is a rhombus.
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Not sure I understand what you are trying to say.

Anyway:
From (1) we have that the parallelogram is also a rhombus (because the sides are equal).
From (2) we have that the parallelogram is also a rectangle (because the diagonals are equal).

So, our parallelogram is a rhombus AND a rectangle, so it's a square!
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I have a doubt in the explanation of this question. The official ans says that all four sides of parallelogram ABCD are equal, which implies that ABCD is a rhombus but this is the property of square(a parallelogram) as well...?\

HI Bunnel,

Diagonal of a square is also equals. then if both the diagonals are equal and root 2 then we have side as 1 and we can calculate the area.

Please clarify.

Please read the red part in my solution. Why should the sides equal to 1? Why cannot they be any numbers satisfying \(x^2+y^2=2\)?
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For more on this trap check the following questions:
the-circular-base-of-an-above-ground-swimming-pool-lies-in-a-167645.html
figure-abcd-is-a-rectangle-with-sides-of-length-x-centimete-48899.html
in-right-triangle-abc-bc-is-the-hypotenuse-if-bc-is-13-and-163591.html
m22-73309-20.html
points-a-b-and-c-lie-on-a-circle-of-radius-1-what-is-the-84423.html
if-vertices-of-a-triangle-have-coordinates-2-2-3-2-and-82159-20.html
if-p-is-the-perimeter-of-rectangle-q-what-is-the-value-of-p-135832.html
if-the-diagonal-of-rectangle-z-is-d-and-the-perimeter-of-104205.html
what-is-the-area-of-rectangular-region-r-105414.html
what-is-the-perimeter-of-rectangle-r-96381.html
pythagorean-triples-131161.html
given-that-abcd-is-a-rectangle-is-the-area-of-triangle-abe-127051.html
m13-q5-69732-20.html#p1176059
m20-07-triangle-inside-a-circle-71559.html
what-is-the-perimeter-of-rectangle-r-96381.html
what-is-the-area-of-rectangular-region-r-166186.html
if-distinct-points-a-b-c-and-d-form-a-right-triangle-abc-129328.html

Hope this helps.
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Because diagonal of a square = site root2

now as it is given diagonals are equal and this is also property of a square . so if diagonal is root 2 then my site will be 1. and area of a square would be one.

Thanks

First of all from (2) we know that ABCD is a rectangle, not necessarily a square.

Next, the fact that the diagonals equals to \(\sqrt{2}\) does not mean that the sides must be equal to 1. The sides can be:

\(\frac{1}{2}\) and \(\frac{\sqrt{7}}{2}\);
\(\frac{1}{3}\) and \(\frac{\sqrt{7}}{\sqrt{3}}\);
...

Basically the lengths of the sides can be any positive (x, y) satisfying \(x^2+y^2=(\sqrt{2})^2\).

Please follow the links in my post above for questions which use the same trap.
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Dear Bunuel, can we say - based on statement 2- that the parallelogram could be rhombus? If the answer is not can

you tell me why?

From (2) we have that ABCD is a rectangle, and if it's a square, then it becomes a rhombus too.
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I would really appreciate if a fault in my logic is pointed out.

Statement 1: ABCD is either a square or a rhombus, so different areas. Insufficient.

Statement 2: ABCD is a parallelogram with equal diagonals, so cannot be a rhombus. Possibly a rectangle or a square. Insufficient.

1+2. It must be a square.

Answer C

Hii
if we know one of the diagonal length & we know that it is a rhombus ( as all sides are equal)
We can find the length of both diagonals using geometrical properties and also the area. (Tag me if you want to know the procedure)
Area of rhombus = (product of diagonal/2)

Answer would be C in that case also.


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Hi

the area of rhombus is not fixed only by the length of sides of rhombus.
To find are of rhombus, following information required:
1) Both diagonals
2) sides and angle between sides.
3) sides and a diagonal

In fact Square has the max area of all the rhombus having same side lengths.



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