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If p is the perimeter of rectangle Q, what is the value of p?

Question: \(P=2(a+b)=?\)

(1) Each diagonal of rectangle Q has length 10. \(d^2=a^2+b^2=100\). Not sufficient. (2) The area of rectangle Q is 48. \(ab=48\). Not sufficient.

(1)+(2) Square P --> \(P^2=4(a^2+b^2+2ab)\). Now as from (1) \(a^2+b^2=100\) and from (2) \(ab=48\), then \(P^2=4(a^2+b^2+2ab)=4(100+2*48)=4*196\) --> \(P=\sqrt{4*196}=2*14=28\). Sufficient.

If instead, Q were a square, would 1 be sufficient?

In a rectangle, why can't we use the Isosceles Triangle to figure out the third side since the diagonals bisect each other?

If we were told that Q is a square instead of a rectangle, then the answer would be D.

As for the second question: can you please explain what you mean? Generally you cannot find the sides of a rectangle just knowing the length of its diagonal, since knowing the length of hypotenuse (diagonal) in a right triangle (created by length and width), is not enough to find the legs of it (length and width).

Re: If p is the perimeter of rectangle Q, what is the value of p [#permalink]

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12 Dec 2012, 14:40

I'm sorry I'm still not seeing how this is not answer "A". I understand the logic at arriving at answer "C", I just don't understand why you NEED to combine statements "1" and "2", contradicts my entire understanding of Data Sufficiency logic.

A rectangle is comprised of 4 right angles, no?

So ultimately the "diagonal" represents the hypotenuse forming two right triangles, no?

Can you form a right triangle with a hypotenuse of 10 with any other legs besides 6 and 8? Or do I have that wrong?

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13 Dec 2012, 03:21

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kelleygrad05 wrote:

I'm sorry I'm still not seeing how this is not answer "A". I understand the logic at arriving at answer "C", I just don't understand why you NEED to combine statements "1" and "2", contradicts my entire understanding of Data Sufficiency logic.

A rectangle is comprised of 4 right angles, no?

So ultimately the "diagonal" represents the hypotenuse forming two right triangles, no?

Can you form a right triangle with a hypotenuse of 10 with any other legs besides 6 and 8? Or do I have that wrong?

(pythagorean triplet (3, 4, 5) , (6, 8, 10))

Hi Kellygrad05,

There was a similar problem I was attempting yesterday on the forum.

Basically we are told that it is a rectangle but we aren't sure if the sides are Integers or not. For ex.

Diagonal-10, sides can be 6 and 8 (because of PT) or something like Square root 99 and 1...and such other combination

When you consider the st2 with above then we can figure out sides will be 6 and 8 as only in that condition Area will be 48 and Diagonal as 10.

Thanks
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I'm sorry I'm still not seeing how this is not answer "A". I understand the logic at arriving at answer "C", I just don't understand why you NEED to combine statements "1" and "2", contradicts my entire understanding of Data Sufficiency logic.

A rectangle is comprised of 4 right angles, no?

So ultimately the "diagonal" represents the hypotenuse forming two right triangles, no?

Can you form a right triangle with a hypotenuse of 10 with any other legs besides 6 and 8? Or do I have that wrong?

(pythagorean triplet (3, 4, 5) , (6, 8, 10))

A right triangle with hypotenuse 10, doesn't mean that we have (6, 8, 10) right triangle. If we are told that the lengths of all sides are integers, then yes: the only integer solution for right triangle with hypotenuse 10 would be (6, 8, 10).

To check this: consider the right triangle with hypotenuse 10 inscribed in circle. We know that a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s side, then that triangle is a right triangle.

So ANY point on circumference of a circle with diameter of 10 would make the right triangle with diameter. Not necessarily sides to be 6 and 8. For example we can have isosceles right triangle, which would be 45-45-90: and the sides would be \(\frac{10}{\sqrt{2}}\). OR if we have 30-60-90 triangle and hypotenuse is \(10\), sides would be \(5\) and \(5*\sqrt{3}\). Of course there could be many other combinations.

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13 Dec 2012, 08:37

Thank you Bunuel, it's clear my understanding of pythagorean triplets was incomplete. The example of the triangle within the circle was quite illuminating. So to summarize, if it is given that all sides of the triangle are integers, and the hypotenuse was given, only then I could have deduced it was part of a pythagorean triple, correct? Was that my only misstep at arriving at answer "A"?

Thank you Bunuel, it's clear my understanding of pythagorean triplets was incomplete. The example of the triangle within the circle was quite illuminating. So to summarize, if it is given that all sides of the triangle are integers, and the hypotenuse was given, only then I could have deduced it was part of a pythagorean triple, correct? Was that my only misstep at arriving at answer "A"?

Re: If p is the perimeter of rectangle Q, what is the value of p [#permalink]

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17 Dec 2013, 12:07

Bunuel wrote:

If p is the perimeter of rectangle Q, what is the value of p?

(1) Each diagonal of rectangle Q has length 10. (2) The area of rectangle Q is 48.

Diagnostic Test Question: 48 Page: 26 Difficulty: 650

1) Basically, in saying that the diagonal is 10, they are giving us the hypotenuse of a right triangle. There is no info about the two other sides though, so insufficient.

2) They are simply telling us the area, which is not enough for us to know the perimeter since there are many different products of two that can yield 48.

However, taking 1 and 2 together, they are giving us the hypothenuse (in 1) and the RELATION between the two other sides of the triangle (in statement 2). Since the pythagoran theorem restricts which size two sides can have, if we are given the third (hypothenuse), then this relation between the other two sides is enough.

Notice that I did not do any calculation at all. The DS questions are more about "does this make sense?" than they are about testing if exact boundaries and relations hold up.

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Re: If p is the perimeter of rectangle Q, what is the value of p [#permalink]

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18 Mar 2015, 21:46

I'm was baffled at how the answer wasn't A as well, since when applying the 30-60-90 x, \sqrt{3} , and 2x you could technically get the other sides. We know the hypotenuse is 10, so we have 2x = 10, so x would be 5 and the last side would be 5\sqrt{3}...

But the second statement contradicts this I guess... something to look out for! I thought I was being clever applying that concept.

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16 May 2015, 08:54

Hi,

I am surprised as well that A was not the correct answer but not for the reasons explained in the previous posts (except if I missed something).

The question is stating that we have a rectangle to consider.

1) tells us that each diagonal of rectangle Q has length 10.

I would guess a rectangle that has its diagonals equal is always a square. If this is a square then knowing the hypotenuse (the diagonal) is enough to guess the perimeter.

I am surprised as well that A was not the correct answer but not for the reasons explained in the previous posts (except if I missed something).

The question is stating that we have a rectangle to consider.

1) tells us that each diagonal of rectangle Q has length 10.

I would guess a rectangle that has its diagonals equal is always a square. If this is a square then knowing the hypotenuse (the diagonal) is enough to guess the perimeter.

Anyone to help me on this? Thanks

The diagonals of a rectangle are always equal.
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Re: If p is the perimeter of rectangle Q, what is the value of p [#permalink]

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16 May 2015, 09:29

Bunuel wrote:

tsunagaru wrote:

Hi,

I am surprised as well that A was not the correct answer but not for the reasons explained in the previous posts (except if I missed something).

The question is stating that we have a rectangle to consider.

1) tells us that each diagonal of rectangle Q has length 10.

I would guess a rectangle that has its diagonals equal is always a square. If this is a square then knowing the hypotenuse (the diagonal) is enough to guess the perimeter.

Anyone to help me on this? Thanks

The diagonals of a rectangle are always equal.

Indeed... I have been a bit quick in my guess. Thanks a lot!

Re: If p is the perimeter of rectangle Q, what is the value of p [#permalink]

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21 Jul 2015, 15:05

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I did not use the math way this is how I did it

it is given that 2L +2w= the perimeter of a rectangle 1. each diagnal of a rectangle is length of 10 which makes the rectangle in half so it cant, but just know the triangle height = not sufficient 2. the area of a rectangle is 48 so l*w= area not sufficent

both will tell us 2 equations and can find the length an width to get p so it is C thanks

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