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

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

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16 Jul 2012, 03:56
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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

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Re: If p is the perimeter of rectangle 0, what is the value of p  [#permalink]

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20 Jul 2012, 03:40
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SOLUTION

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.

Similar questions to practice:
if-the-diagonal-of-rectangle-z-is-d-and-the-perimeter-of-104205.html
what-is-the-area-of-rectangular-region-r-105414.html
what-is-the-perimeter-of-rectangle-r-96381.html

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Re: If p is the perimeter of rectangle 0, what is the value of p  [#permalink]

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16 Jul 2012, 10:00
4
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Let x and y be two sides of the rectangle.. then p =x+y

Stat1
Diagonal = 10
=> x^2 + y^2 = 10^2
we cannot find p i.e. x+y using this info so NOT SUFFICIENT

Sta2
Area = 48
=> xy =48
we cannot find p i.e. x+y using this info so NOT SUFFICIENT

Combining (1) and (2)
we will get
value of x+y = sqrt (x+y)^2 = sqrt(x^2 + y^2 + 2xy) = sqrt ( 10^2 + 2*48)
sqrt(196) = 14

SUFFICIENT
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Re: If p is the perimeter of rectangle 0, what is the value of p  [#permalink]

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28 Jul 2012, 14:42
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?
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Re: If p is the perimeter of rectangle 0, what is the value of p  [#permalink]

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29 Jul 2012, 00:36
ctiger100 wrote:
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).

Hope it's clear.
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Re: If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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12 Dec 2012, 13:40
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?

(pythagorean triplet (3, 4, 5) , (6, 8, 10))
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Re: If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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13 Dec 2012, 02:21
3
1
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))

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

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13 Dec 2012, 03:18
6
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.

Similar questions to practice:
if-the-diagonal-of-rectangle-z-is-d-and-the-perimeter-of-104205.html
what-is-the-area-of-rectangular-region-r-105414.html
what-is-the-perimeter-of-rectangle-r-96381.html

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

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13 Dec 2012, 07:37
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"?
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Re: If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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13 Dec 2012, 07:43
1
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"?

Yes, that's correct.

For more check Triangles chapter of out Math Book: math-triangles-87197.html

DS geometry questions: search.php?search_id=tag&tag_id=32
PS geometry questions: search.php?search_id=tag&tag_id=53

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

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17 Dec 2013, 11:07
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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16 May 2015, 07:54
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.

Anyone to help me on this?
Thanks
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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, 08:10
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.
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Re: If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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21 Jul 2015, 14:05
2
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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Re: If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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04 Aug 2015, 12:17
1
While solving this problem , I thought that the diagonal must split the rectangle in 2 Triangles 30,60,90° - But then found this rule
A diagonal splits a rectangle into two congruent 30-60-90 triangles, if, and only if, the diagonal is twice as long as the width of the rectangle
May be it helps some of you.
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Re: If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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04 Aug 2015, 13:01
BrainLab wrote:
While solving this problem , I thought that the diagonal must split the rectangle in 2 Triangles 30,60,90° - But then found this rule
A diagonal splits a rectangle into two congruent 30-60-90 triangles, if, and only if, the diagonal is twice as long as the width of the rectangle
May be it helps some of you.

A quick way to remember this is to know that all squares are rectangles but not all rectangles are square. Additionally, diagonals of a square bisect the 90 degree angles. Thus the 2 triangles formed by any 1 of the diagonals has angles 45-45-90. So this shows that the rectangle can have different angles and not just 30-60-90 or 45-45-90. The actual values will depend upon the values of either one of l or b or length of the diagonal (we will need 2 of them).
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If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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04 Aug 2015, 13:15
Thanks Engr2012. I have already repeated my Geometry script.. Actually we can be just sure that we have a right triangle, but other information cannot be deducted from the statement (as 30,60,90°), and as you said we have only more information about a diagonal of a square.
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OG Diagnostic Q48 (PS, Geometry)  [#permalink]

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29 Nov 2015, 06:24
From the Diagnostic test:

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

The first fact states that each diagonal of Q has length 10, which I thought was sufficient. The answer key has C as the answer.

My reasoning: If the diagonal is 10 and this is a rectangle, then the diagonal makes right triangles. The other sides must be 6 and 8 from the special property of a 3-4-5 right triangle. With the sides I can find the perimeter.

The explanation goes on to solve for the sides using both facts (#2 is that the area is 48). The book comes to w=6 and l=8, as I expected from #1 alone.

How is statement one NOT sufficient? Is there any way for the rectangle to have a diagonal of 10 but NOT have sides 6 and 8? What am I missing here?
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Re: OG Diagnostic Q48 (PS, Geometry)  [#permalink]

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29 Nov 2015, 06:30
jz4analytics wrote:
From the Diagnostic test:

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

The first fact states that each diagonal of Q has length 10, which I thought was sufficient. The answer key has C as the answer.

My reasoning: If the diagonal is 10 and this is a rectangle, then the diagonal makes right triangles. The other sides must be 6 and 8 from the special property of a 3-4-5 right triangle. With the sides I can find the perimeter.

The explanation goes on to solve for the sides using both facts (#2 is that the area is 48). The book comes to w=6 and l=8, as I expected from #1 alone.

How is statement one NOT sufficient? Is there any way for the rectangle to have a diagonal of 10 but NOT have sides 6 and 8? What am I missing here?

Hi,
what you are missing is that you are considering the sides to be integer values..
it can be non integer...
for example $$5\sqrt{2}$$ and $$5\sqrt{2}$$...
there can be various other combinations..
hope it helped
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Re: If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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29 Nov 2015, 06:36
jz4analytics wrote:
From the Diagnostic test:

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

The first fact states that each diagonal of Q has length 10, which I thought was sufficient. The answer key has C as the answer.

My reasoning: If the diagonal is 10 and this is a rectangle, then the diagonal makes right triangles. The other sides must be 6 and 8 from the special property of a 3-4-5 right triangle. With the sides I can find the perimeter.

The explanation goes on to solve for the sides using both facts (#2 is that the area is 48). The book comes to w=6 and l=8, as I expected from #1 alone.

How is statement one NOT sufficient? Is there any way for the rectangle to have a diagonal of 10 but NOT have sides 6 and 8? What am I missing here?

Follow posting guidelines (link in my signatures), search for a question before you post. This question has already been discussed before. Whenever you are talking about a new question, make sure to quote the complete questions and options.

As for your question, diagonal of 10 can be obtained by sides $$\sqrt {99}$$ and $$1$$, $$5\sqrt{3}$$ and $$5$$ etc. Nowhere it is mentioned that the sides are integers.
Re: If p is the perimeter of rectangle Q, what is the value of p &nbs [#permalink] 29 Nov 2015, 06:36

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