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If the area of a parallelogram is 100, what is the perimeter
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KarishmaB
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Re: If the area of a parallelogram is 100, what is the perimeter [#permalink] New post  21 Sep 2016, 22:13
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rohit8865 wrote:
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:
The attachment PointLatticeParallelograms_1000.gif is no longer available

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:
The attachment m10-31.png is no longer available


Hope it's clear.

Bunuel/Anyone
Why not A alone is sufficient
being known 2 sides are 10 it means angles are also same .also as height BD is making 90 deg. means the other 2 angles are 45 deg each
fig. attached)
Thus making A as ans choice

Thanks


Responding to a pm:

Height is the length of the altitude from a vertex to the opposite side. It will be 10 in this case. But why is it necessary that it will fall on D? Look at these two figures. The perimeter in each case will be different.

Attachment:
ParallelogramJPEG.jpg
ParallelogramJPEG.jpg [ 23.09 KiB | Viewed 2348 times ]

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Re: If the area of a parallelogram is 100, what is the perimeter [#permalink] New post  13 Oct 2019, 02:58
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.


We know from 2 that AD=BD, hence and hence we can find out base of the //gm , and we can find of the other side using pythagoras on sin45. hence finding the perimeter. I am unable to understand that how is B insufficient?
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Re: If the area of a parallelogram is 100, what is the perimeter [#permalink] New post  10 Jul 2020, 05:15
rush123 wrote:
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.


We know from 2 that AD=BD, hence and hence we can find out base of the //gm , and we can find of the other side using pythagoras on sin45. hence finding the perimeter. I am unable to understand that how is B insufficient?

Because we don't have any specific value of Base and Height. 100 area can have many possible values (100, 1) (25,4) (20,5) ... And so on. That's why it's insufficient.

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Re: If the area of a parallelogram is 100, what is the perimeter [#permalink] New post  09 Mar 2021, 22:14
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.
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Applicant4991
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If the area of a parallelogram is 100, what is the perimeter [#permalink] New post  29 Aug 2021, 22:00
Hi Everyone,

I did try to understand all the other explanations.

However I am still not able to solve a scenario based on statement II wherein if we assume angle ABC to be 45 degrees. (Considering any angle can be 45)

In case of the above assumption I suppose we cannot get a 45-45-90 triangle.

Please let me know what is the flaw in my above assumption.
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Re: If the area of a parallelogram is 100, what is the perimeter [#permalink] New post  30 Aug 2021, 04:15
Applicant4991 wrote:
Hi Everyone,

I did try to understand all the other explanations.

However I am still not able to solve a scenario based on statement II wherein if we assume angle ABC to be 45 degrees. (Considering any angle can be 45)

In case of the above assumption I suppose we cannot get a 45-45-90 triangle.

Please let me know what is the flaw in my above assumption.

From statement 1. It's given that the height of parallelogram is 10 (since base is 10) but we can't determine the length of slant height of parallelogram without knowing anything about the angles.
A is not sufficient.

Statement B states one of the angle of parallelogram is 45°
Since there are two equal acute angles in parallelogram and 2 equal obtuse angles. Here B and D are acute angles of 45°
You can see and analyze which angle will be acute of obtuse for this.
Back to question.
The length of base could be 100 and height 1 or vice versa or both could be 10 each.
There are 3 possible answers here.
Not sufficient.

Adding both statements together,
We get this diagram type structure.
Pardon for my screen scribbling. It's definitely not good. But I hope you get the diagram.

We can analyse the slant height of parallelogram using right angle triangle property.
That's why answer is C

Hope this helps.

Posted from my mobile device
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Re: If the area of a parallelogram is 100, what is the perimeter [#permalink] New post  03 Dec 2021, 02:11
https://youtu.be/fOXFwnW1_2U

here is a video answer for anyone interested! :)
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Re: If the area of a parallelogram is 100, what is the perimeter [#permalink] New post  17 Mar 2022, 10:32
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.


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Re: If the area of a parallelogram is 100, what is the perimeter [#permalink]
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