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

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Intern
Joined: 27 Jul 2010
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If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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05 Oct 2010, 01:11
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Difficulty:

25% (medium)

Question Stats:

77% (01:10) correct 23% (01:42) wrong based on 207 sessions

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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.

Official Guide 12 Question

 Question: 48 Page: 26 Difficulty: 750

Find All Official Guide Questions

Video Explanations:
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Joined: 02 Sep 2009
Posts: 50572

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05 Oct 2010, 01:27
2
wininblue 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

I am not able to understand how do you solve a quadratic eqn of this size and conclude that length and breadth can have 2 distinct values.

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

(1) $$d^2=a^2+b^2=100$$. Not sufficient.
(2) $$ab=48$$. 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 $$P^2=4(a^2+b^2+2ab)=4(100+2*48)=4*196$$ --> $$P=\sqrt{4*196}=28$$. Sufficient.

So you can see that it's not necessary to solve quadratic equation for a and b to get P.

Similar question: one-more-geometry-96381.html?hilit=perimeter%20rectangle#p742145

Hope it helps.
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05 Oct 2010, 02:57
Great Method!
Thank you so much Bunuel, +1
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05 Oct 2010, 03:23
Since it is a rectangle, the diagnal opposite side would make 90 degrees, so the diagnal = a^2 + b^2 which is equal to 10 ------ from (i)

I can nt determine the p value from the (i)

From (2), ab = 48 not sufficient. now combine both, we know that (a+b)^2 = a^2 + b^2 + 2ab .... we have all the values ..

so the ans is : C
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06 Oct 2010, 06:20
Perimeter = p = 2* (l+b)

To caluculate the perimeter, we need both l and b

1) Length of diagonal = 10
Length of a diagonal = sqrt{l^2 + b^2} = 10
Squaring both sides, we get (l^2 + b^2) = 100
This is not sufficient since we have one equation and 2 variables.

2) Area = l*b = 48
This is not sufficient since we have one equation and 2 variables.

(l+b)^2 = l^2 + b^2 + 2*l*b
Substituting the values obtained in 1) and 2), we can get the value of (l+b) and thus we can calculate the perimeter.

Hence C.
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Perimeter of rectangle Q ?  [#permalink]

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30 Jan 2011, 07:14
Hi,

in the book "GMAT review 12th edt.", there is diagnostic test question #48 (DS).
-----
If p is the perimeter of rectangle Q, what is the value of p?
1) Each diagonal of rectangle Q has length of 10.
2) The area of rectangle Q is 48.
----
Now, the answer explanation says C is correct. However, when looking at answer 1), I know the hypotenuse of both triangles is 10. Using the Pythagorean theorem, I know that my sides are 8 and 6 -> (5:4:3) x 2.

So p = 2l + 2w = 16 + 12... hence A is sufficient to determine the value.

Where is my error?
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Posts: 50572
Re: Perimeter of rectangle Q ?  [#permalink]

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30 Jan 2011, 07:33
1
Merging similar topics.

demeuse81 wrote:
Hi,

in the book "GMAT review 12th edt.", there is diagnostic test question #48 (DS).
-----
If p is the perimeter of rectangle Q, what is the value of p?
1) Each diagonal of rectangle Q has length of 10.
2) The area of rectangle Q is 48.
----
Now, the answer explanation says C is correct. However, when looking at answer 1), I know the hypotenuse of both triangles is 10. Using the Pythagorean theorem, I know that my sides are 8 and 6 -> (5:4:3) x 2.

So p = 2l + 2w = 16 + 12... hence A is sufficient to determine the value.

Where is my error?

Check this post: one-more-geometry-96381.html#p742164

You assume with no ground for it that the lengths of the sides are integers. Knowing that hypotenuse equals to 10 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 6:8:10. Or in other words: if $$a^2+b^2=10^2$$ DOES NOT mean that $$a=6$$ and $$b=8$$, certainly this is one of the possibilities but definitely not the only one. In fact $$a^2+b^2=10^2$$ has infinitely many solutions for $$a$$ and $$b$$ and only one of them is $$a=6$$ and $$b=8$$.

For example: $$a=1$$ and $$b=\sqrt{99}$$ or $$a=2$$ and $$b=\sqrt{96}$$ or $$a=4$$ and $$b=\sqrt{84}$$ ...

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.

Check this post for more on triangles: math-triangles-87197.html

Hope it helps.
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30 Jan 2011, 09:27
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

For the rectangle; if
Diagonal=d
length=l
width=w

Rephrasing: What is 2(l+w) or simply what is (l+w)

1) d=10

$$l^2+w^2=d^2$$
$$(l+w)^2-2lw=d^2$$
$$(l+w)^2=d^2+2lw=10^2+2lw=100+2lw$$
$$(l+w)=sqrt{100+2lw}$$

If we knew lw; we would have known l+w; but we don't know that
Not Sufficient.

2) Area=48
lw=48
$$l^2+w^2=d^2$$
$$(l+w)^2-2lw=d^2$$
$$(l+w)^2=d^2+2lw=d^2+2*48=d^2+96$$
$$(l+w)=sqrt{d^2+96}$$

If we knew diagonal "d"; we would have known l+w; but we don't know that
Not Sufficient.

Using both the statements;
We know lw=48 and d=10

$$l^2+w^2=d^2$$
$$(l+w)^2-2lw=d^2$$
$$(l+w)^2=d^2+2lw=10^2+2*48=100+96$$
$$(l+w)=sqrt{196}=14$$

Sufficient using both statements.

Ans: "C"
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Official Guide - 12 - Question D48  [#permalink]

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14 Sep 2011, 23:13
Hi.. am a member for some while now, however this is my 1st ever post. In Q 48 (data sufficiency) of diagnostic test of OG-12, the diagonal length of rectangle is given as 10 inches, and we need to find the perimeter.
According to me, the statement is sufficient in that the sides have to be 6 and 8 inhes (using the pythagorean triple 6-8-10). However, book says it's not sufficient.

Can someone plz clarify or explain? Tx
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Re: Official Guide - 12 - Question D48  [#permalink]

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17 Sep 2011, 00:05
deephoenix wrote:
Hi.. am a member for some while now, however this is my 1st ever post. In Q 48 (data sufficiency) of diagnostic test of OG-12, the diagonal length of rectangle is given as 10 inches, and we need to find the perimeter.
According to me, the statement is sufficient in that the sides have to be 6 and 8 inhes (using the pythagorean triple 6-8-10). However, book says it's not sufficient.

Can someone plz clarify or explain? Tx

OG is right. You cannot take 6-8-10 to solve this quesion. In the above posts Bunnel and fluke have explained the solution of this question using the right approach. Please refer their posts and reply back if you have any doubts.
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Re: If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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09 Dec 2017, 05:47
i chose answer A since this is a rectangle and the diagonal will bisect the right angles, forming 45-45-90 triangle with sides ratios of 1:1:root 2.
using Pythagorean theory , we can get the value of both sides

what did i do wrong here ?
Math Expert
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Posts: 50572
If p is the perimeter of rectangle Q, what is the value of p  [#permalink]

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09 Dec 2017, 05:54
mohamk wrote:
i chose answer A since this is a rectangle and the diagonal will bisect the right angles, forming 45-45-90 triangle with sides ratios of 1:1:root 2.
using Pythagorean theory , we can get the value of both sides

what did i do wrong here ?

The diagonals of a rectangle bisect the angle if and only the rectangle is a square. Generally, a diagonal of a rectangle can form any angle with the adjacent sides from 0 to 90, not inclusive.

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-p-is-the- ... 35832.html In case of any question please post in that open topic.
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If p is the perimeter of rectangle Q, what is the value of p &nbs [#permalink] 09 Dec 2017, 05:54
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