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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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Updated on: 20 Jul 2012, 03:32
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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

Originally posted by zisis on 03 Nov 2010, 15:10.
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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Re: The Quest for 700: Weekly GMAT Challenge  [#permalink]

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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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Re: The Quest for 700: Weekly GMAT Challenge  [#permalink]

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

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Updated on: 10 Jan 2013, 00:18
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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

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

Originally posted by mbaiseasy on 12 Dec 2012, 20:11.
Last edited by mbaiseasy on 10 Jan 2013, 00:18, edited 1 time in total.
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Re: If the diagonal of rectangle Z is d, and the perimeter of  [#permalink]

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

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

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

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24 Feb 2019, 12:32
I made it into a 3-4-5 triangle:
diagonal=5
side = 3,4 so P = 14 and target Area is 3*4=12

Then plug into answers, E) comes out to be (196-100)/8 = 96/8 = 12
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Re: If the diagonal of rectangle Z is d, and the perimeter of  [#permalink]

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18 Mar 2020, 10:10
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Re: If the diagonal of rectangle Z is d, and the perimeter of   [#permalink] 18 Mar 2020, 10:10