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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]  26 Nov 2010, 12:55
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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.
[Reveal] Spoiler: OA

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Re: NEED SOME Help on this DS question [#permalink]  26 Nov 2010, 13:07
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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.

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Re: NEED SOME Help on this DS question [#permalink]  27 Nov 2010, 09:03
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]  27 Nov 2010, 09:22
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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]  27 Nov 2010, 13:13
I feel so dumb now, I never thought about it.

Thanks for the clarification!
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QR: DS 42 Geomatry [#permalink]  20 Feb 2011, 12:33
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

OG solution is very long.
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Last edited by Baten80 on 20 Feb 2011, 12:47, edited 1 time in total.
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Re: What is the area of rectangular region R? [#permalink]  29 Feb 2012, 01:30
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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]  29 Feb 2012, 01:40
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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]  29 Feb 2012, 03:32
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]  20 Apr 2012, 19:05
Almost fell for the 3,4,5 rule, good explanation.
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Re: NEED SOME Help on this DS question [#permalink]  21 Sep 2012, 08:15
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.

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.

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Re: NEED SOME Help on this DS question [#permalink]  21 Sep 2012, 08:23
Expert's post
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.

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.

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]  08 Dec 2012, 10:06
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]  09 Dec 2012, 08:49
Expert's post
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]  11 Apr 2014, 00:04
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.

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]  11 Apr 2014, 01:35
Expert's post
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.

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]  11 Apr 2014, 02:29
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.

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]  22 Jul 2014, 07:33
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]  22 Jul 2014, 07:36
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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] 22 Jul 2014, 07:36
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