M13-05

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.

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.

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

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

Thanks

Not necessarily, it might also be a rhombus.

Check for more here: math-polygons-87336.html

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

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?

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.

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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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.

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.