Rectangular park

[alex.gellatly]

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

[Anurag@Gurome]

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 = √(a² + b²) = 200 ---> (a² + b²) = 200²

Therefore, area = ab = [(a + b)² - (a² + b²)]/2 = [280² - 200²]/2 = (280 - 200)(280 + 200)/2 = (80*480)/2 = 40*480 = 19200

The correct answer is A.

[Lifetron]

A, It is !

[alex.gellatly]

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 = √(a² + b²) = 200 ---> (a² + b²) = 200²

Therefore, area = ab = [(a + b)² - (a² + b²)]/2 = [280² - 200²]/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

[eagleeye]

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 = √(a² + b²) = 200 ---> (a² + b²) = 200²

Therefore, area = ab = [(a + b)² - (a² + b²)]/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)]

[truplayer256]

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 = x² + y² = 200²

Area = xy = 1/2*[(x + y)² - (x² + y²)] = 1/2* [280² - 200²] = (480)(80)/2 = 480(40) = 19200

Choose A.

[armand_h]

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.