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 Post subject: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:31 am 
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Is the area of the top of a rectangular pool larger than 1000 square feet?
(1) The pool's top measures 50 feet diagonally.
(2) One side of the pool's top measures 25 feet.

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.

(C) Statement (1) tells us the diagonal measure, but we don't know the dimensions. The pool's top might be very long and narrow, measuring less than a foot on one side. In this case its area would be well under 1000 square feet. The pool could also be 30 by 40 feet, with a diagonal of 50 feet and an area of 1200 square feet. So we can't answer whether the area of the pool is greater than 1000 square feet.

Statement (2) tells us the length of one side, but we need the length and the width of the pool to determine its area.

Combining the two statements, we could use the Pythagorean theorem to determine the missing dimension of the pool. We could then determine whether the area is larger than 1000 square feet. This is a Data Sufficiency question, so you should not waste time with the calculation. Since combining the statements gives us sufficiency, the correct answer is C.

Note: If you want to determine the actual area of the pool, you can use the properties of triangles to quickly calculate the area. Since the length of one side of the pool is half the length of the diagonal, we know that the triangle formed by the two sides of the pool and the diagonal is a 30-60-90 triangle. The side that is 25 feet must be the short side, and the longer side will be √3 times the length of the short side, or 25√3 feet. The area is thus 25 × 25√3 = 625√3, which is larger than 1000 square feet.
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Not clear how rectangle can be long and thin. When I see a diagonal of 50, I think of a 3:4:5 Triangle. In the explanation provided, you show a length of 49 as one side and height of 1 as the other. However, when applying Pythagorean square root of 1² + 49² equals a hypotenuse of 49 not 50 as shown in the graph.

Is there an alternative explanation to this question?


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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:33 am 
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Let us analyze the statement (1) first, using the Pythagorean theorem.
The only value it defines is the diagonal, which equals 50 ft.

The rectangle can be:
Image
The area is 30 × 40 = 1200 > 1000 ft².

Or the rectangle can be:
Image
The area is 1 × √(50² – 1²) < 1 × 50 = 50 < 1000 ft²

Let us analyze the statement (2).
The only value it defines is one of the sides, which equals 25 ft.

The other side can be 1:
Image
The area is 1 × 25 = 25 < 1000 ft²

The other side can be 100:
Image
The area is 100 × 25 = 2500 > 1000 ft²

Therefore each statement by itself is NOT sufficient.

When we use the both statements together, the triangle (1/2 of the rectangle) is the following:
Image

Using the Pythagorean theorem the third side is √(50² – 25²).
The area of the rectangle is 25 × √(50² – 25²), which is a definite value that is either greater or less than 1000 ft².
Therefore the both statements together are sufficient to answer the question.

P.S. 25 × √(50² – 25²) = 25 × √1875 < 25 × √1600 = 25 × 40 = 1000.
So 25 × √(50² – 25²) < 1000 ft²
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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:37 am 
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great explanation.

Thank You.


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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:38 am 
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Why A is not sufficient? Isn't it right triangle with sides 50 40 30? If Hypotenuse is 50 doesn't it mean that other sides are 30 and 40 for sure?


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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:39 am 
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Quote:
If Hypotenuse is 50 doesn't it mean that other sides are 30 and 40 for sure?
No. If two sides of the right triangle were set, that would be enough. (The third side would be defined by Pythagorean Theorem).
If a hypotenuse is the only defined side then there are infinitely many various right triangles that have the same hypotenuse. Here is the example of such triangles:
Image
(The third vertex of such triangles is situated on the circle, which has this hypotenuse as a diameter).
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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:39 am 
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Hello. I don´t understand why the answer is C, because your explanation does not use the information stated in Statement 1, this is, where and how did you use the 50 feet of the diagonal in the approach? I see my mistake, but please help me see why is it important, cause I only see that you used the 25ft to solve by the Pythagorean theorem.

Thank you very much!


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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:40 am 
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questioner wrote:
Hello. I don´t understand why the answer is C, because your explanation does not use the information stated in Statement 1, this is, where and how did you use the 50 feet of the diagonal in the approach? ..., cause I only see that you used the 25ft to solve by the Pythagorean theorem.
In order to use Pythagorean theorem here we need to know two sides of the right triangle.
Image
One side is given by statement (2), it is 25ft. The other side (hypotenuse) is given by statement (1), it is 50 ft.
The third side of the triangle (it is also a side of the rectangle) can be found from Pythagorean theorem:
50² = 25² + x²
Note, that 50² = (2 × 25)² = 4 × 25². So
4 × 25² = 25² + x²
3 × 25² = x²
25√3 = x.
The desired side is 25√3 ft.


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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:40 am 
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Statement 1 is sufficient to answer the question.
l² + b² = 2500,
l² – 2500 + b² = 0.
We can break 2500 as 2lb, so lb = 1250, which is the area of the rectangular top.


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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:41 am 
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questioner wrote:
We can break 2500 as 2lb
It is an assumption. It is NOT based on any facts given in the question statement and statement (1).

This assumption is the same as if you assume that the sides of the pool are equal:

l² – 2500 + b² = 0.
assuming that 2500 = 2lb leads to
l² – 2lb + b² = 0
(lb)² = 0
l = b

But the sides of the pool are NOT necessarily equal. For example, one side can be 3 times longer than the other one. In this case l = 3b

So l² + b² = 2500 transforms into
(3b)² + b² = 2500
9b² + b² = 2500
10b² = 2500
b² = 250
b = √250
b = 5√10
l = 15√10
In this case 2lb = 2 × 5√10 × 15√10 = 2 × 750 = 1500. (Not 2500, how it is in the case l = b)

There are infinitively many such triangles that have a hypotenuse 50 (AB = 50):
Image


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 Post subject: Re: GMAT Geometry
PostPosted: Wed Apr 10, 2013 2:41 am 
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Got it, thanks!


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