Rectangular park

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alex.gellatly: Master | Next Rank: 500 Posts; Posts: 435; Joined: Wed Nov 16, 2011 7:27 amFollowed by: 16 members; Thanked: 48 times

by alex.gellatly » Tue Jul 17, 2012 9:57 pm

A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

19200
19600
20000
20400
20800

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Source: Beat The GMAT — Problem Solving | Tue Jul 17, 2012 9:57 pm

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Tue Jul 17, 2012 10:12 pm

alex.gellatly wrote:A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

Say, the lengths of the sides of the rectangle are a and b in feet.
Hence, perimeter = 2(a + b) = 560 ---> (a + b) = 280
And, length of diagonal = âˆš(aÂ² + bÂ²) = 200 ---> (aÂ² + bÂ²) = 200Â²

Therefore, area = ab = [(a + b)Â² - (aÂ² + bÂ²)]/2 = [280Â² - 200Â²]/2 = (280 - 200)(280 + 200)/2 = (80*480)/2 = 40*480 = 19200

The correct answer is A.

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Lifetron: Master | Next Rank: 500 Posts; Posts: 308; Joined: Thu Mar 29, 2012 12:51 amFollowed by: 3 members; Thanked: 16 times

by Lifetron » Sat Jul 21, 2012 8:01 am

A, It is !

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alex.gellatly: Master | Next Rank: 500 Posts; Posts: 435; Joined: Wed Nov 16, 2011 7:27 amFollowed by: 16 members; Thanked: 48 times

by alex.gellatly » Wed Jul 25, 2012 1:31 am

Anurag@Gurome wrote:
alex.gellatly wrote:A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?
Say, the lengths of the sides of the rectangle are a and b in feet.
Hence, perimeter = 2(a + b) = 560 ---> (a + b) = 280
And, length of diagonal = âˆš(aÂ² + bÂ²) = 200 ---> (aÂ² + bÂ²) = 200Â²

Therefore, area = ab = [(a + b)Â² - (aÂ² + bÂ²)]/2 = [280Â² - 200Â²]/2 = (280 - 200)(280 + 200)/2 = (80*480)/2 = 40*480 = 19200I get confused about here!

The correct answer is A.

Thank you very much, but I get confused along the way. Can you explain the bold part above?
Thanks

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eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Wed Jul 25, 2012 2:21 am

alex.gellatly wrote:
Anurag@Gurome wrote:
alex.gellatly wrote:A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?
Say, the lengths of the sides of the rectangle are a and b in feet.
Hence, perimeter = 2(a + b) = 560 ---> (a + b) = 280
And, length of diagonal = âˆš(aÂ² + bÂ²) = 200 ---> (aÂ² + bÂ²) = 200Â²

Therefore, area = ab = [(a + b)Â² - (aÂ² + bÂ²)]/2

Thank you very much, but I get confused along the way. Can you explain the bold part above?
Thanks
We know that
(a+b)^2 = a^2 + b^2 +2ab = (a^2 + b^2) + 2ab
=> (a+b)^2 - (a^2 + b^2) = 2ab
=> 2ab = (a+b)^2 - (a^2 + b^2)
=> ab = 1/2 [ (a+b)^2 - (a^2 + b^2)]

We know that area of a rectangle = length*width = a*b = ab.
So area = ab = 1/2 [ (a+b)^2 - (a^2 + b^2)]

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truplayer256: Master | Next Rank: 500 Posts; Posts: 392; Joined: Thu Jan 15, 2009 12:52 pm; Location: New Jersey; Thanked: 76 times

by truplayer256 » Wed Jul 25, 2012 7:05 pm

Let the length of the park be x and let the width be y. It then follows that:

P = Perimeter = 2x + 2y = 560 => x + y = 280

From the pythagorean theorem:

Diagonal = xÂ² + yÂ² = 200Â²

Area = xy = 1/2*[(x + y)Â² - (xÂ² + yÂ²)] = 1/2* [280Â² - 200Â²] = (480)(80)/2 = 480(40) = 19200

Choose A.

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armand_h: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Wed Jul 25, 2012 9:02 am; Thanked: 1 times

by armand_h » Thu Jul 26, 2012 1:21 am

to simplify things, let's divide a and b by 10

A small, rectangular park has a perimeter of 56 feet and a diagonal measurement of 20 feet. What is its area, in square feet?

192
196
200
204
208

The answer is either 192 or 196:
The maximum surface is when it's a square, surface= diagonal^2 / 2 =20*20/2=200

We know that a+b is an even number and a*b is also an even number => a and b should both be even.
If the answer is 196, then the only combination with a and b being even numbers is 98 and 2, which doesn't match the perimeter (98+2)*2=200
The only answer left is 192.

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