Bunuel wrote:
mirzohidjon wrote:
Hi, Bunuel,
I really appreciate your effort to help us. Finally after your explanation, I got the answer (since I imagined what kind of parallelogram the problem was talking about.
But, based on what assumptions, you deduce that parallelogram needs to be the way you described (diagonal equal to height)?
Attachment:
PointLatticeParallelograms_1000.gif
If the area of a parallelogram is 100, what is the perimeter of the parallelogram?
Given: \(Area=base*height=100\). Q: \(P=2b+2l=?\) (b - base, l - leg )
(1) The base of the parallelogram is 10 --> \(base=height=10\). Infinite variations are possible. Look at the diagram (let the distance between two horizontal and vertical points be 10): all 4 parallelograms have \(base=height=10\) but they have different perimeter. Not sufficient. Side notes: \(l\geq{10}\), when \(l=10=h\) we would have the square (case #3 on the diagram) and \(P=40\) (smallest possible perimeter), maximum value of perimeter is not limited.
(2) One of the angles of the parallelogram is 45 degrees. Clearly insufficient. But from this statement height BX and AX will make isosceles right triangle: \(height=BX=AX\).
(1)+(2) As from 2 we have that \(height=BX=AX\) and from (1) we have that \(base=height=10\) --> \(AX=base=AD=10\) --> X and D coincide (case #4 on the diagram) --> leg (AB) becomes hypotenuse of the isosceles right triangle with sides equal to 10 --> \(AB=10\sqrt{2}\) --> \(P=20+20\sqrt{2}\). Sufficient.
Answer: C.
Attachment:
m10-31.png
Hope it's clear.
Hi
Bunuel,
This is an amazing question. I have now understood the solution.
I got it wrong (as E) because I got it to the point that there could be two values for the angle, say angle A of the parallelogram, 45 degrees and 135 degrees. And as such there are two different/distinct parallelograms formed. So, together, these two statements are still not sufficient to find a unique answer/perimeter.
After I solved with your method, I got to know that only 1 type of parallelogram is possible. As you have shown and we can find it's perimeter, so correct answer is C.
I solved again, by taking angle A as 45 degrees and finding it's perimeter. Then again, taking angle A as 135 degrees and finding it's perimeter. And in both the cases that I thought, it's the same perimeter coming after solving.
Can you suggest that when should we stop or check further as in such type of questions?
Do we need to draw the figure and verify that both parallelograms are effectively same.
TIA.
Regards,
Ravish.