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What is the area of rectangular region R ? [#permalink]
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26 Nov 2010, 13:55
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What is the area of rectangular region R ? (1) Each diagonal of R has length 5. (2) The perimeter of R is 14.
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Re: NEED SOME Help on this DS question [#permalink]
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ajit257 wrote: What is the area of rectangular region R ? (1) Each diagonal of R has length 5 (2) The perimeter of R is 14
Please could someone explain this question ...thanks. Let the sides of the rectangle be \(x\) and \(y\). Question: \(area=xy=?\) (1) Each diagonal of R has length 5 > as the diagonals in a rectangle are the hypotenuses for the sides then: \(x^2+y^2=5^2\), but we can not get the value of \(xy\) from this info. Not sufficient. (2) The perimeter of R is 14 > \(P=2(x+y)=14\) > \(x+y=7\). Again we can not get the value of \(xy\) from this info. Not sufficient. (1)+(2) We have \(x^2+y^2=25\) and \(x+y=7\). Square the second expression: \(x^2+2xy+y^2=49\), as \(x^2+y^2=5^2\) then \(25+2xy=49\) > \(xy=12\). Sufficient. Answer: C.
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Re: NEED SOME Help on this DS question [#permalink]
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27 Nov 2010, 10:03
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I thought "Statement 1" alone is sufficient to solve this problem.
3,4,5 is the only Pythagorean triplet which supports 5 a diagonal of a right angled triangle.
Why can't the answer be A?



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Re: NEED SOME Help on this DS question [#permalink]
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27 Nov 2010, 10:22
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rockroars wrote: I thought "Statement 1" alone is sufficient to solve this problem.
3,4,5 is the only Pythagorean triplet which supports 5 a diagonal of a right angled triangle.
Why can't the answer be A? We are not told that the lengths of the sides are integers. So knowing that hypotenuse equals to 5 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple  3:4:5. Or in other words: if \(x^2+y^2=5^2\) DOES NOT mean that \(x=3\) and \(y=4\). Certainly this is one of the possibilities but definitely not the only one. In fact \(x^2+y^2=5^2\) has infinitely many solutions for \(x\) and \(y\) and only one of them is \(x=3\) and \(y=4\). For example: \(x=1\) and \(y=\sqrt{24}\) or \(x=2\) and \(y=\sqrt{21}\)... So knowing that the diagonal of a rectangle (hypotenuse) equals to one of the Pythagorean triple hypotenuse value is not sufficient to calculate the sides of this rectangle. Hope it's clear.
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Re: NEED SOME Help on this DS question [#permalink]
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27 Nov 2010, 14:13
I feel so dumb now, I never thought about it. Thanks for the clarification!



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QR: DS 42 Geomatry [#permalink]
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20 Feb 2011, 13:33
QR: DS 42 what is the area of rectangualr region R? (1) Each diagonal R has length 5 (1) The perimeter of R is 14 I solved as follows: L=x, W= y, D= z x^2+y^2=z^2 1. z^2=25 N.S 2. 2(x+y)=14 => x+y=7 => (x+y)^2=49 (doing square both side) N.S for are 2xy ito be calculated so, x^2+y^2=z^2 => (x+y)^2  2xy=z^2 => 49 2xy=25 2xy=24 Is my approach correct? Please help. OG solution is very long.
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Last edited by Baten80 on 20 Feb 2011, 13:47, edited 1 time in total.



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Re: What is the area of rectangular region R? [#permalink]
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29 Feb 2012, 02:30
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From what I understand of rectangular diagonals or quadrilateral diagonals is that if they are the same length, then all sides should be of equal length. Also area of Rhombus = 1/2 * diagonal * diagonal? Correct me if I'm wrong here, just need clarification



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Re: What is the area of rectangular region R? [#permalink]
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29 Feb 2012, 02:40



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Re: What is the area of rectangular region R ? (1) Each diagonal [#permalink]
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29 Feb 2012, 04:32
Actually I just realized it, sounded so stupid. Thanks!



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Re: What is the area of rectangular region R ? (1) Each diagonal [#permalink]
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20 Apr 2012, 20:05
Almost fell for the 3,4,5 rule, good explanation.



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Re: NEED SOME Help on this DS question [#permalink]
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21 Sep 2012, 09:15
Bunuel wrote: ajit257 wrote: What is the area of rectangular region R ? (1) Each diagonal of R has length 5 (2) The perimeter of R is 14
Please could someone explain this question ...thanks. Let the sides of the rectangle be \(x\) and \(y\). Question: \(area=xy=?\) (1) Each diagonal of R has length 5 > as the diagonals in a rectangle are the hypotenuses for the sides then: \(x^2+y^2=5^2\), but we can not get the value of \(xy\) from this info. Not sufficient. (2) The perimeter of R is 14 > \(P=2(x+y)=14\) > \(x+y=7\). Again we can not get the value of \(xy\) from this info. Not sufficient. (1)+(2) We have \(x^2+y^2=25\) and \(x+y=7\). Square the second expression: \(x^2+2xy+y^2=49\), as \(x^2+y^2=5^2\) then \(25+2xy=49\) > \(xy=12\). Sufficient. Answer: C. Area of a rectangular region = Product of two diagonals/2 We are given both are diagonals are equal to 5 So area would be = 25/2 = 12.5 Thus A is sufficient Let me know why i am wrong. Waiting for reply.
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Re: NEED SOME Help on this DS question [#permalink]
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21 Sep 2012, 09:23
fameatop wrote: Bunuel wrote: ajit257 wrote: What is the area of rectangular region R ? (1) Each diagonal of R has length 5 (2) The perimeter of R is 14
Please could someone explain this question ...thanks. Let the sides of the rectangle be \(x\) and \(y\). Question: \(area=xy=?\) (1) Each diagonal of R has length 5 > as the diagonals in a rectangle are the hypotenuses for the sides then: \(x^2+y^2=5^2\), but we can not get the value of \(xy\) from this info. Not sufficient. (2) The perimeter of R is 14 > \(P=2(x+y)=14\) > \(x+y=7\). Again we can not get the value of \(xy\) from this info. Not sufficient. (1)+(2) We have \(x^2+y^2=25\) and \(x+y=7\). Square the second expression: \(x^2+2xy+y^2=49\), as \(x^2+y^2=5^2\) then \(25+2xy=49\) > \(xy=12\). Sufficient. Answer: C. Area of a rectangular region = Product of two diagonals/2 We are given both are diagonals are equal to 5 So area would be = 25/2 = 12.5 Thus A is sufficient Let me know why i am wrong. Waiting for reply. The red part is not correct. It's true about squares: \(area_{square}=\frac{diagonal^2}{2}\). Hope it helps.
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Re: What is the area of rectangular region R ? [#permalink]
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08 Dec 2012, 11:06
I screwed up on this one like an earlier poster. So hypothetically, if the question stem stated that the sides were integers, would A be sufficient alone? I'm nervous on the DS. I got one wrong on the PS in the official guide and 6 wrong already on DS and I'm only on question 50 .



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Re: What is the area of rectangular region R ? [#permalink]
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09 Dec 2012, 09:49



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Re: NEED SOME Help on this DS question [#permalink]
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11 Apr 2014, 01:04
Bunuel wrote: ajit257 wrote: What is the area of rectangular region R ? (1) Each diagonal of R has length 5 (2) The perimeter of R is 14
Please could someone explain this question ...thanks. Let the sides of the rectangle be \(x\) and \(y\). Question: \(area=xy=?\) (1) Each diagonal of R has length 5 > as the diagonals in a rectangle are the hypotenuses for the sides then: \(x^2+y^2=5^2\), but we can not get the value of \(xy\) from this info. Not sufficient. (2) The perimeter of R is 14 > \(P=2(x+y)=14\) > \(x+y=7\). Again we can not get the value of \(xy\) from this info. Not sufficient. (1)+(2) We have \(x^2+y^2=25\) and \(x+y=7\). Square the second expression: \(x^2+2xy+y^2=49\), as \(x^2+y^2=5^2\) then \(25+2xy=49\) > \(xy=12\). Sufficient. Answer: C. Hi Bunuel, Can't we apply the 1 sqrt3 2 theory to statement one? Since it's a rectangular then the angle created by the diagonal must be 90 and leaving the rest 30 and 60. So the sides must be 5/2 and 5/2(sqrt3). What's wrong with this explanation? Thanks.



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Re: NEED SOME Help on this DS question [#permalink]
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11 Apr 2014, 02:35
aquax wrote: Bunuel wrote: ajit257 wrote: What is the area of rectangular region R ? (1) Each diagonal of R has length 5 (2) The perimeter of R is 14
Please could someone explain this question ...thanks. Let the sides of the rectangle be \(x\) and \(y\). Question: \(area=xy=?\) (1) Each diagonal of R has length 5 > as the diagonals in a rectangle are the hypotenuses for the sides then: \(x^2+y^2=5^2\), but we can not get the value of \(xy\) from this info. Not sufficient. (2) The perimeter of R is 14 > \(P=2(x+y)=14\) > \(x+y=7\). Again we can not get the value of \(xy\) from this info. Not sufficient. (1)+(2) We have \(x^2+y^2=25\) and \(x+y=7\). Square the second expression: \(x^2+2xy+y^2=49\), as \(x^2+y^2=5^2\) then \(25+2xy=49\) > \(xy=12\). Sufficient. Answer: C. Hi Bunuel, Can't we apply the 1 sqrt3 2 theory to statement one? Since it's a rectangular then the angle created by the diagonal must be 90 and leaving the rest 30 and 60. So the sides must be 5/2 and 5/2(sqrt3). What's wrong with this explanation? Thanks. Let me ask you a question: why must the remaining angles be 30 and 60 degrees? Why cannot they be 25 or 65? Or 20 and 70? Basically any pair totaling 90?
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Re: NEED SOME Help on this DS question [#permalink]
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11 Apr 2014, 03:29
aquax wrote: Bunuel wrote: ajit257 wrote: What is the area of rectangular region R ? (1) Each diagonal of R has length 5 (2) The perimeter of R is 14
Please could someone explain this question ...thanks. Let the sides of the rectangle be \(x\) and \(y\). Question: \(area=xy=?\) (1) Each diagonal of R has length 5 > as the diagonals in a rectangle are the hypotenuses for the sides then: \(x^2+y^2=5^2\), but we can not get the value of \(xy\) from this info. Not sufficient. (2) The perimeter of R is 14 > \(P=2(x+y)=14\) > \(x+y=7\). Again we can not get the value of \(xy\) from this info. Not sufficient. (1)+(2) We have \(x^2+y^2=25\) and \(x+y=7\). Square the second expression: \(x^2+2xy+y^2=49\), as \(x^2+y^2=5^2\) then \(25+2xy=49\) > \(xy=12\). Sufficient. Answer: C. Hi Bunuel, Can't we apply the 1 sqrt3 2 theory to statement one? Since it's a rectangular then the angle created by the diagonal must be 90 and leaving the rest 30 and 60. So the sides must be 5/2 and 5/2(sqrt3). What's wrong with this explanation? Thanks. Probably you are getting confused because of this example: "a rectangle is inscribed in a circle of radius r...." The diagonal divides the rectangle in two right triangles, so sum of two angles need to be 90 but it can be 30: 60, 45:45...and so on...



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Re: What is the area of rectangular region R ? [#permalink]
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22 Jul 2014, 08:33
Hi Bunuel,
To keep it straight, just because it says its a rectangle does not mean we have to have two 306090 triangles, but if we put together two 306090 triangles we get a rectangle? Correct? I picked "A" because I thought that since it said we had a rectangle, we had to have two of these triangles. From the discussion above, it looks like this is not a mandatory condition of a rectangle.



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Re: What is the area of rectangular region R ? [#permalink]
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Re: What is the area of rectangular region R ? [#permalink]
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07 Aug 2017, 16:52
Can someone please explain why we can't make four triangles out of the rectangle to determine the third side of each triangle?




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