M10-31

Question

If the area of a parallelogram is 100 square units, what is its perimeter?

(1) The parallelogram has a base measuring 10 units.
 
(2) One of the parallelogram's angles measures 45 degrees.

Bunuel · Accepted Answer

Official Solution:

If the area of a parallelogram is 100 square units, what is its perimeter? 
 
Refer to the diagram below:
 
 https://gmatclub.com/gmat-practice-tests/resources/import/img/m10-31.png 
 
Given:  Area = base*height = 100 . We want to find the perimeter,  P = 2b + 2l , where  b  is the base and  l  is the leg (side length).
 
(1) The parallelogram has a base measuring 10 units.
 
So,  base = height = 10 . Infinitely many cases are possible. Observe the diagram (let the distance between two horizontal and vertical points be 10 units): all 4 parallelograms have  base = height = 10  but they have different perimeters. Therefore, this information is not sufficient.
 
(2) One of the angles of the parallelogram is 45 degrees. 
 
Clearly insufficient by itself. However, from this statement, we know that height  BX  and  AX  form an isosceles right triangle:  height = BX = AX .
 
(1) + (2) Combining both statements, from (2) we have that  height = BX = AX , and from (1) we have that  base = height = 10 , so  AX = base = AD = 10 . This means that points  X  and  D  coincide (case #4 in the diagram). Thus, leg  AB  becomes the hypotenuse of the isosceles right triangle with sides equal to 10 units, hence  AB = 10\sqrt{2} . Therefore, the perimeter  P = 20 + 20\sqrt{2} . Both statements together are sufficient.

Answer: C

rohan0510 · Answer

Why can't I use the second statement to calculate the length of the other side using sin 45. I know the height is 10. Shouldn't the answer be B?

Bunuel · Answer

How do you know that height is 10 for (2)?

kishor1 · Answer

Good explanation. Thank you.

Posted from my mobile device

TheMBmonster · Answer

I think could be b) for the following reason:

We know from the stem that the area is 100, and from stmt 2 that one of the angles of the parallelogram is 45 degrees.

We know that opposite angles are equal : so we have 2 degree angles, so the other angles have to be 135 degrees

Is there not only 1 such parallelogram that has an area of 100 degrees?

Thus the lengths sides could be induced, and thus the perimeter calculated?

Bunuel · Answer

I addressed that doubt here: https://gmatclub.com/forum/if-the-area- ... ml#p737846

mbeneduzi · Answer

Hi  Bunuel  . Could you answer what I've missed in my reasoning?

If one side of a parallelogram has 45 degrees, we actually have 2 sides of 45 degrees and 2 sides of 135 degrees. Going further, we can make 4 45-45-90 triangles sharing the same height as one side of the triangles, thus each triangle area = 25. Once you know that each triangle has an area = 25 and 2 sides of each are equal, we could find the sides and the hypotenuse, thus the perimeter would be 2*(2*side + hypotenuse), making B the correct answer. What is wrong with that?

anshulpal15 · Answer

See, the problem lies here. You can make 2 45-45-90 triangles but not 4 because when you drop two perpendiculars(which are heights actually), you do not know if they are making equal intercepts as the heights. Or in other words, by dropping two perpendiculars(which are heights), the internal figure thus obtained would be rectangle not square, therefore 45-45-90 degs triangle is not possible. Everything, apart from that was right.
I hope you got my point.

JayeshIBM · Answer

Area of parallelogram =absin@ =100 
 from 1 we get a=10 , therefore b=10√2 
 hence both statement together are sufficient

sssanskaar · Answer

understood why option A is insufficient

- we know that area of ||gm = b x h = 100
- from St1, we know that b = 10 so we know that h is also 10.

Plugging in the values of perimeter = 2(b+l), we can NOT get the value of perimeter. Hence, St1 is insufficient.

MHIKER · Answer

The area of parallelogram = base * height

(1) If the base (a side of the parallelogram) 10 then the height is 10 too.

(2) Is insufficient.

considering both:
The 45-degree angle creates a 45-45-90 right triangle, where aside (hypotenuse). of the parallelogram is  10\sqrt{2} .
Now we have all the sides of the parallelogram to determine its perimeter. Sufficient.

The answer is C

ishitaagarwal59 · Answer

okay so AX= BX = 10. but why is AX=AD?

AnirudhaS · Answer

because AD=10 (this is statement 1)
So, AX=AD=10,
and therefore X and D coincides

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.

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