If the diagonal of rectangle Z is d, and the perimeter of

Question

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

Bunuel · Accepted Answer

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} .

Answer: E.

KarishmaB · Answer

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.

zisis · Answer

excellent explanation ! kudos

mbaiseasy · Answer

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

Cassiss · Answer

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

sambam · Answer

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.

KarishmaB · Answer

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.

PareshGmat · Answer

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}

Answer = E

hsbinfy · Answer

the question rotating on the formula:
(x+y)^2=x^2+y^2+2xy

just solve acc and get answer E

energetics · Answer

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

jppresa · Answer

Could be solved by just assuming the rectangle is a square with sides of 2 or 3?

888iceman888 · Answer

Is this a 650 level question?

Bunuel · Answer

You can check difficulty level of a question along with the stats on it in the first post. For this question Difficulty = 600-700 Level. The difficulty level of a question is calculated automatically based on the timer stats from the users which attempted the question.

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If the diagonal of rectangle Z is d, and the perimeter of

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