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What is the area of rectangular region R ?

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What is the area of rectangular region R ?  [#permalink]

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New post 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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New post 26 Nov 2010, 14:07
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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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New post 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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New post 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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New post 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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New post Updated on: 20 Feb 2011, 13:47
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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Originally posted by Baten80 on 20 Feb 2011, 13:33.
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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New post 29 Feb 2012, 02:30
1
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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New post 29 Feb 2012, 02:40
1
calvin1984 wrote:
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


All rectangles have the diagonals of equal length, so (1) doesn't necessarily means that given rectangle is a rhombus.

For more on this subject check Polygons chapter of Math Book: math-polygons-87336.html

Hope it helps.
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Re: What is the area of rectangular region R ? (1) Each diagonal  [#permalink]

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New post 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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New post 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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New post 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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New post 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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New post 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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New post 09 Dec 2012, 09:49
RonBagel wrote:
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?


Yes, if we were told that the lengths of the sides of the rectangle are integers, then the first statement would be sufficient: x^2+y^2=25 --> x=3 and y=4 or vise -versa --> xy=12.
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Re: NEED SOME Help on this DS question  [#permalink]

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New post 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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New post 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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New post 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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New post 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 30-60-90 triangles, but if we put together two 30-60-90 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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New post 22 Jul 2014, 08:36
1
jbdoyl3 wrote:
Hi Bunuel,

To keep it straight, just because it says its a rectangle does not mean we have to have two 30-60-90 triangles, but if we put together two 30-60-90 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.


Correct. But you can get a rectangle by putting together any two congruent right triangles, not necessarily 30-60-90 triangles.
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Re: What is the area of rectangular region R ?  [#permalink]

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New post 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?
Re: What is the area of rectangular region R ? &nbs [#permalink] 07 Aug 2017, 16:52

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