If the diagonal of rectangle Z is d, and the perimeter of : GMAT Problem Solving (PS)
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# If the diagonal of rectangle Z is d, and the perimeter of

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If the diagonal of rectangle Z is d, and the perimeter of [#permalink]

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03 Nov 2010, 15:10
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If the diagonal of rectangle Z is d, and the perimeter of rectangle Z is p, what is the area of rectangle Z, in terms of d and p?

(A) (d^2 – p)/3
(B) (2d^2 – p)/2
(C) (p – d^2)/2
(D) (12d^2 – p^2)/8
(E) (p^2 – 4d^2)/8
[Reveal] Spoiler: OA

Last edited by Bunuel on 20 Jul 2012, 03:32, edited 1 time in total.
Edited the question and added the OA.
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03 Nov 2010, 15:17
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zisis wrote:
If the diagonal of rectangle Z is d, and the perimeter of rectangle Z is p, what is the area of rectangle Z, in terms of d and p?
(A) (d2 – p)/3
(B) (2d2 – p)/2
(C) (p – d2)/2
(D) (12d2 – p2)/8
(E) (p2 – 4d2)/8

NO OA - once provided i will update

IMO (A)

Let the sides of rectangle be $$x$$ and $$y$$.

Given: $$d^2=x^2+y^2$$ and $$p=2(x+y)$$. Question: $$area=xy=?$$

Square $$p$$ --> $$p^2=4(x^2+2xy+y^2)$$ --> substitute $$x^2+y^2$$ by $$d^2$$ --> $$p^2=4(d^2+2xy)$$ --> $$p^2-4d^2=8xy$$ --> $$area=xy=\frac{p^2-4d^2}{8}$$.

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03 Nov 2010, 15:55
Bunuel wrote:
zisis wrote:
If the diagonal of rectangle Z is d, and the perimeter of rectangle Z is p, what is the area of rectangle Z, in terms of d and p?
(A) (d2 – p)/3
(B) (2d2 – p)/2
(C) (p – d2)/2
(D) (12d2 – p2)/8
(E) (p2 – 4d2)/8

NO OA - once provided i will update

IMO (A)

Let the sides of rectangle be $$x$$ and $$y$$.

Given: $$d^2=x^2+y^2$$ and $$p=2(x+y)$$. Question: $$area=xy=?$$

Square $$p$$ --> $$p^2=4(x^2+2xy+y^2)$$ --> substitute $$x^2+y^2$$ by $$d^2$$ --> $$p^2=4(d^2+2xy)$$ --> $$p^2-4d^2=8xy$$ --> $$area=xy=\frac{p^2-4d^2}{8}$$.

excellent explanation ! kudos
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Re: The Quest for 700: Weekly GMAT Challenge [#permalink]

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03 Nov 2010, 18:47
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When answer is in terms of 1 or 2 variables, my suggestion would be to quickly take simple values. (If variables are more than that, keeping a track of their values becomes cumbersome)
I would say let the sides be 3 and 4 to get diagonal, d = 5 (Pythagorean triple). Then p = 2x3 + 2x4 = 14. You are looking for an area of 3x4 = 12
Put values and check.

There is a teeny-weeny chance that two options may give you the answer you are looking for. In that case, you might have to take different values and put in those two options to pick out the winner, but the risk is worth it, in my opinion.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 13 Aug 2012 Posts: 464 Concentration: Marketing, Finance GMAT 1: Q V0 GPA: 3.23 Followers: 25 Kudos [?]: 433 [0], given: 11 Re: If the diagonal of rectangle Z is d, and the perimeter of [#permalink] ### Show Tags 12 Dec 2012, 20:11 Equation 1: $$d = \sqrt{L^2+W^2}$$ Equation 2: $$p = 2(L+W)$$ Combine 1 and 2: $$d^2 = L^2+W^2$$ $$p^2 = 4 ( L^2 + 2LW + W^2)$$ $$p^2 = 4L^2 + 8LW + 4W^2$$ $$p^2 = 4d^2 + 8LW$$ Remember that A = LW and \frac{p}{2}=L+W $$A = \frac{p^2-4d^2}{8}$$ Answer: E _________________ Impossible is nothing to God. Last edited by mbaiseasy on 10 Jan 2013, 00:18, edited 1 time in total. Intern Joined: 20 Dec 2012 Posts: 2 Location: France Concentration: International Business, Strategy Schools: LBS '14 WE: Project Management (Telecommunications) Followers: 0 Kudos [?]: 6 [0], given: 3 Re: If the diagonal of rectangle Z is d, and the perimeter of [#permalink] ### Show Tags 09 Jan 2013, 09:39 if we take a rectangle of 2x1 area A = 2x1 = 2 perimeter P = 2x2 + 2x1 = 6 the diagonal D = sqrt(1²+2²) = sqrt(5) for ansewer (B) (2d^2 – p)/2 (2D² - P) = 2*sqrt(5)² - 6 = 4 ==> 4/2 = 2 ==> which is the area A the same reasoning goes for answer E am i doing something wrong? because B and E are correct for me Manager Joined: 18 Oct 2011 Posts: 90 Location: United States Concentration: Entrepreneurship, Marketing GMAT Date: 01-30-2013 GPA: 3.3 Followers: 2 Kudos [?]: 69 [0], given: 0 Re: If the diagonal of rectangle Z is d, and the perimeter of [#permalink] ### Show Tags 09 Jan 2013, 10:56 I just used real numbers to solve this question. Pick a rectangle with sides 3 and 4. Diagonal would be 5 so would = d. Perimeter = 14, Area = 12. Only E works. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7125 Location: Pune, India Followers: 2137 Kudos [?]: 13673 [0], given: 222 Re: If the diagonal of rectangle Z is d, and the perimeter of [#permalink] ### Show Tags 09 Jan 2013, 19:58 Cassiss wrote: if we take a rectangle of 2x1 area A = 2x1 = 2 perimeter P = 2x2 + 2x1 = 6 the diagonal D = sqrt(1²+2²) = sqrt(5) for ansewer (B) (2d^2 – p)/2 (2D² - P) = 2*sqrt(5)² - 6 = 4 ==> 4/2 = 2 ==> which is the area A the same reasoning goes for answer E am i doing something wrong? because B and E are correct for me When you take some numbers as you did here (say, distinct sides are 2 and 1 so p = 6, d = ...etc), it is possible that 2 or more options give you the correct answer. In that case, you need to take another set of numbers and put the values in only those two options. Hopefully, only one of them will give you the correct answer. Also, when picking numbers, try to pick those which need minimum effort. e.g. I would like to pick the sides as 3 and 4 because I know that the diagonal in that case will be 5. No messy square roots to handle. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: If the diagonal of rectangle Z is d, and the perimeter of [#permalink]

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15 Jul 2014, 12:54
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Re: If the diagonal of rectangle Z is d, and the perimeter of [#permalink]

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15 Jul 2014, 21:20
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Refer diagram below:

Let one side of the rectangle = a

Other side $$= \frac{p}{2} - a$$

Require to find $$Area = a (\frac{p}{2} - a) = \frac{ap}{2} - a^2$$

$$d^2 = a^2 + (\frac{p}{2} - a)^2$$

$$d^2 = 2a^2 + \frac{p^2}{4} - ap$$

$$ap - 2a^2 = \frac{p^2}{4} - d^2$$

$$\frac{ap}{2} - a^2 = \frac{p^2 - 4d^2}{8}$$

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Re: If the diagonal of rectangle Z is d, and the perimeter of [#permalink]

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24 Jul 2015, 12:42
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25 Jul 2016, 07:21
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Re: If the diagonal of rectangle Z is d, and the perimeter of [#permalink]

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26 Jul 2016, 00:55
the question rotating on the formula:
(x+y)^2=x^2+y^2+2xy

just solve acc and get answer E
Re: If the diagonal of rectangle Z is d, and the perimeter of   [#permalink] 26 Jul 2016, 00:55
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