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sanjoy18: Senior | Next Rank: 100 Posts; Posts: 81; Joined: Tue Jun 11, 2013 10:24 pmFollowed by: 1 member; Thanked: 7 times

by sanjoy18 » Sat Oct 04, 2014 7:38 am

Square ABCD has an area of 9 square inches. Sides AD and BC are lengthened to x inches each. By how many inches were sides AD and BC lengthened?

(1) The diagonal of the resulting rectangle measures 5 inches.
(2) The resulting rectangle can be cut into three rectangles of equal size.

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Source: Beat The GMAT — Data Sufficiency | Sat Oct 04, 2014 7:38 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Oct 04, 2014 7:57 am

sanjoy18 wrote:Square ABCD has an area of 9 square inches. Sides AD and BC are lengthened to x inches each. By how many inches were sides AD and BC lengthened?

(1) The diagonal of the resulting rectangle measures 5 inches.
(2) The resulting rectangle can be cut into three rectangles of equal size.

Since the area of square ABCD = 9, AB=CD=AD=BC = 3.

Statement 1: The diagonal of the resulting rectangle measures 5 inches.
Implication:
AD=BC=4, so that the diagonal of the resulting rectangle is the hypotenuse of a 3-4-5 triangle.
Thus, AD and BC are each lengthened by x=1 inch.
SUFFICIENT.

Statement 2: The resulting rectangle can be cut into three rectangles of equal size.
ANY rectangle can be divided into 3 rectangles of equal size.
To divide a rectangle into 3 rectangles of equal size, simply draw through the interior 2 parallel lines that split the rectangle into 3 equal smaller rectangles.
Since the resulting triangle in statement 2 can be of ANY SIZE, the value of x cannot be determined.
INSUFFICIENT.

The correct answer is A.

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by [email protected] » Sun Oct 05, 2014 1:18 pm

Hi sanjoy18,

Mitch's explanation showcases a Geometry rule that you're likely to see on Test Day: the 3/4/5 right triangle. The GMAT is a Test of pattern-based questions, so you should be on the lookout for patterns (and each subset of the Quant and Verbal sections has its own built-in patterns).

When it comes to Geometry, right triangles show up in a number of situations. Here, by cutting the square/rectangle from corner-to-corner, we end up with 2 right triangles. When a right triangle is given (or implied), you should think about the Pythagorean Theorem, the 3/4/5 and 5/12/13 right triangles and the 30/60/90 and 45/45/90 triangles. Chances are pretty good that one or more of those patterns is built into the question.

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Anaira Mitch: Master | Next Rank: 500 Posts; Posts: 235; Joined: Wed Oct 26, 2016 9:21 pmFollowed by: 5 members; Thanked: 3 times

by Anaira Mitch » Sat Dec 31, 2016 2:43 am

If the square has an area of 9 square inches, it must have sides of 3 inches each. Therefore, sides AD and BC have lengths of 3 inches each. These sides are lengthened to x inches, while the other two remain at 3 inches. This gives us a rectangle with two opposite sides of length x and two opposite sides of length 3. Then we are asked by how much the two lengthened sides were extended. In other words, what is the value of x - 3? In order to answer this, we need to find the value of x itself.

(1) SUFFICIENT: If the resulting rectangle has a diagonal of 5 inches, we end up with the following:
We can now see that we have a 3-4-5 right triangle, since we have a leg of 3 and a hypotenuse (the diagonal) of 5. The missing leg (in this case, x) must equal 4. Therefore, the two sides were each extended by 4 - 3 = 1 inch.

(2) INSUFFICIENT: It will be possible, no matter what the value of x, to divide the resulting rectangle into three smaller rectangles of equal size. For example, if x = 4, then the area of the rectangle is 12 and we can have three rectangles with an area of 4 each. If x = 5, then the area of the rectangle is 15 and we can have three rectangles with an area of 5 each. So it is not possible to know the value of x from this statement.
The correct answer is A.

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by Jay@ManhattanReview » Mon Jan 02, 2017 1:12 am

sanjoy18 wrote:Square ABCD has an area of 9 square inches. Sides AD and BC are lengthened to x inches each. By how many inches were sides AD and BC lengthened?

(1) The diagonal of the resulting rectangle measures 5 inches.
(2) The resulting rectangle can be cut into three rectangles of equal size.

This question can be solved with a logical deduction approach.

Given that the area of the suqare ABCD = 9 sq inches, we have the side of the square = 3.

S1: Certainly, one can apply a popular template 3-4-5 to get the extended side of the rectangle. However, had statement 1 stated that the diagonal of the resulting rectangle measures [Say 6] inches, even then you could have calculated the value of the extended side of the rectangle by applying Pythogorous theorem. So, indeed there is no need to know what is the value of x till you are sure that x would a unique value and x > 0. Sufficient.

S2: This statement can also be analysed logically. What if the side of the resulting rectangle = 18, in that case x = 18 - 3 = 15. Similarly, if the side of the resulting rectangle = 17 (Any rectangle can be split into three equal rectangles) , in that case x = 17 - 3 = 14. No unique answer. Insufficient.

OA: A

-Jay
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