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Parallelogram

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Parallelogram

by Soumita Ghosh » Sat Jun 23, 2012 5:01 pm
There is a park in a parallelogram shape. Person A walks along the longer diagonal. Person B walks along the shorter one. Does A cover 1.5 times the distance B covers?
1. Parallelogram has one side 1 km and another side 2km
2. One of the internal angles of the parallelogram is 60 degrees
Question is essentially asking if one diagonal is 1.5 times the other diagonal.


can anyone show me how to solve this question !! can anyone tell me what is the easiest formula for finding the diagonals of parallelogram if the length of the two sides of parallelogram are given.

Please help me !!

Thanks a lot in advance !!
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Source: — Data Sufficiency |
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by sandeep_thaparianz » Sun Jun 24, 2012 4:53 am
Statement tells us that two sides are 1 and 2 but if u try to draw a parallelogram with this u can have lot of parallelogram by varying the internal angle. One being a rectangle also(whose diagonals are equal) So statement 1 is insufficient

Statement 2 tells that if we try to draw a parallelogram with one of the angle =60 and other being 120 we can have a number of parallelogram which varying length and breath. So it is also insufficient

Now combine 1 and 2 statement. You will get a only 1 figure of parallelogram.

Hope it helps.
OA is C
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by Soumita Ghosh » Sun Jun 24, 2012 12:34 pm
Hi Sandeep ,

Thanks for your response .

How can you prove that One diagonal is 1.5 times the length of other diagonal ?
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by tutorphd » Sun Jun 24, 2012 1:09 pm
Soumita, in Yes/No data sufficiency problems, you often don't have to calculate, you just have to say if it is possible and whether you can answer the question with a definite Yes or definite No. The problem you asked for is a geometry construction problem - statements (1) and (2) taken together completely fix the parallelogram, you know that by sketching it. The lengths of the two diagonals are fixed and can be determined (you don't need to know how) so that is sufficient to answer the question with a definite yes or no.

If you insist to calculate them, you must know the Cosine Theorem, although GMAT doesnt require trigonometry.

Using the Cosine Theorem:

https://en.wikipedia.org/wiki/Cosine_theorem

The short diagonal is in a triangle with sides 1 and 2 and angle of 60 degrees between them. It's length is d^2 = 1^2 + 2^2 - 2*1*2*cos(60) = 3, so the short diagonal is root(3). This is actually a 30-60-90 triangle. Any triangle with two sides in the ratio 1 : 2 and an angle of 60 degrees between them will be a right triangle of type 30-60-90. The longest side "2" is the hypothenuse.

The long diagonal is in a triangle with sides 1 and 2 and angle of 120 degrees between them. It's length is d^2 = 1^2 + 2^2 - 2*1*2*cos(120) = 7, so the long diagonal is root(7), which is not 1.5 the short diagonal. That answers the question with a definite NO and statements 1 and 2 are sufficient.

Note that answering a Yes/No question with a definite No is still sufficient.
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by Soumita Ghosh » Mon Jun 25, 2012 4:05 pm
Thanks a lot tutorphd for your explanation . Now I have understood it !!
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