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 Post subject: GMAT Geometry
PostPosted: Mon May 13, 2013 9:32 am 
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A piece of carpet in the shape of a trapezoid has a smaller base measuring 50 inches and the larger base measuring 70 inches. What is the perimeter of the carpet?
(1) The area of the carpet is 450 square inches.
(2) The height of the carpet is 7.5 inches.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(E) Statement (1) tells us the area of the piece of carpet. The formula for the area of a trapezoid is:
Area = (0.5(base 1 + base 2) × height).
We can use the given information to solve for the height of the trapezoid:
0.5(70 + 50) × height = 450
60 × height = 450
height = 450/60 = 15/2
height = 7.5.

We cannot determine the perimeter because we know nothing about the lengths of the other two sides. Although you can draw down two perpendiculars to make two right triangles, we don’t know what the base would be for each triangle or if they are the same on both sides of the trapezoid.

In Statement (2), we are given the same information that we solved for in Statement (1). Knowing the height did not allow us to find the perimeter, so it will not help us here either. Statement (2) is insufficient.

Some people might think that you can solve this, but there are an infinite number of possibilities for the lengths of the two sides. Since the statements are both insufficient, and tell us the same thing, combining them will not help us answer the question.

Since the statements are both insufficient, even combined, the correct answer is choice (E).
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I don't get how Statement 1 and Statement 2 are insufficient. Can't you find the length of the two other sides of the trapezoid if the height is found? Could you explain this problem visually?


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 Post subject: Re: GMAT Geometry
PostPosted: Mon May 13, 2013 9:34 am 
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Let's analyse this. What do we know?
the two bases: 50 and 70.
the height: 7.5

So what trapezoids does it define? We can draw two parallel lines 7.5 inches apart:
Image

The bases lie on this lines. We can fix one of the bases (let's fix the larger one) and move another one to imagine all possible trapezoids that fit the requirements:
[Image

We already should understand that the sum of the lengths of sides is not the same for all such trapezoids. But if we do NOT have such understanding and still have doubts, we can take two specific well-known trapezoids:
Image

Draw some heights for easier comprehension:
Image

Then we can easily calculate the sides using Pythagorean theorem:
Image

As wee see, the perimeters in the two cases do differ.
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 Post subject: Re: GMAT Geometry
PostPosted: Mon May 13, 2013 9:34 am 
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We can find the remaining two sides by dropping a perpendicular from top
vertex to the base and similarly dropping another perpendicular to the base.
The three parts formed of the base , will have lengths 10 ,50 & 10. Thus by
Pythagoras theorem we can find the remaining two sides.

Hence, the answer to this is "D".


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 Post subject: Re: GMAT Geometry
PostPosted: Mon May 13, 2013 9:35 am 
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questioner wrote:
The three parts formed of the base , will have lengths 10 ,50 & 10.
The heights from the top vertices will divide the base in the proposed proportion for a very specific trapezoid (on the right).
Image

However there are infinitely many other trapezoids that satisfy the statements.
Image


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 Post subject: Re: GMAT Geometry
PostPosted: Mon May 13, 2013 9:36 am 
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Yep, I too think you are wrong. Knowing the 2 bases and height should be enough because when one of the other sides gets bigger the other gets smaller by the same proportion and the perimeter should remain the same. You asked for the perimeter, not the size of the sides.


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 Post subject: Re: GMAT Geometry
PostPosted: Mon May 13, 2013 9:37 am 
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Quote:
Knowing the 2 bases and height should be enough because when one of the other sides gets bigger the other gets smaller by the same proportion and the perimeter should remain the same.
The area will stay the same, but not the perimeter.

We looked at the example of such two specific trapezoids above.
Image

The perimeter of the trapezoid on the left is about 149.
The perimeter of the trapezoid on the right is 145.
They are not equal.


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