A rectangular park has a perimeter of 340 feet and a diagonal measurement of 130 feet.What is its area, in square feet?

2500

1440

6000

7040

8080

## Area of a Rectangle:

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- GmatMathPro
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The perimeter is 340, so 2x+2y=340 or x+y=170.

Also, from pythagorean theorem, x^2+y^2=130^2

We're looking for the value of x*y, so we want to find a way to get that into the mix. If you remember that (x+y)^2=x^2+2xy+y^2, then you'll see that you can introduce this term by squaring both sides of the perimeter equation:

(x+y)^2=170^2 ---> x^2 + 2xy + y^2 =170^2

Now, subtract the equation x^2+y^2=130 to get 2xy=170^2-130^2

2xy=(170+130)(170-130) (factoring by difference of squares)

2xy=300(40)

2xy=12000

xy=6000

aditya.j wrote:A rectangular park has a perimeter of 340 feet and a diagonal measurement of 130 feet.What is its area, in square feet?

2500

1440

6000

7040

8080

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- Brent@GMATPrepNow
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**Posts:**13576**Joined:**08 Dec 2008**Location:**Vancouver, BC**Thanked**: 5254 times**Followed by:**1256 members**GMAT Score:**770

Let x and y equal the length and width of the rectangle.aditya.j wrote:A rectangular park has a perimeter of 340 feet and a diagonal measurement of 130 feet.What is its area, in square feet?

2500

1440

6000

7040

8080

1. perimeter=340 --> 2x+2y=340 --> x+y=170

2. diagonal=130 --> x^2 + y^2 = 130^2 (from Pythagoras)

Our goal is to find the value of xy (since this equals the area of the rectangle)

If we take x+y=170 and square both sides we get (x+y)^2 = 170^2

If we expand this, we get x^2 + 2xy + y^2 = 170^2

Here's the trick: Notice that we have x^2 + y^2 on the left-hand-side.

From 2, we know that x^2 + y^2 = 130^2

So, we'll take x^2 + 2xy + y^2 = 170^2 and replace x^2 + y^2 with 130^2 to get:

2xy + 130^2 = 170^2

This means that 2xy = 170^2 - 130^2

And this means that xy = (170^2 - 130^2)/2

Which means xy = 6000 = C

Cheers,

Brent

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l+b = 170

Given diagonal = 130. ('l' and 'b' form a right triangle with diagonal 130 and l+b = 170)

*Look for special triangles :*

5-12-13 is a special triangle so 50-120-130 is also a right triangle with 50+120 = 170

therefore l=120 and b=50.

Area = 120*50 = 6000

### GMAT/MBA Expert

- Brent@GMATPrepNow
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Awesome solution, shankar.ashwin!!shankar.ashwin wrote:2 * (l+b) = 340

l+b = 170

Given diagonal = 130. ('l' and 'b' form a right triangle with diagonal 130 and l+b = 170)

Look for special triangles :

5-12-13 is a special triangle so 50-120-130 is also a right triangle with 50+120 = 170

therefore l=120 and b=50.

Area = 120*50 = 6000

I didn't notice the 50-120-130 triangle.

Cheers,

Brent

Brent Hanneson - Creator of GMATPrepNow.com

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We can assume that the other two sides are 50 and 120 when we're told that the diagonal is 130? I see that the dimensions satisfy the perimeter that we're given but, in general, could a triangle with a hypotenuse of 13 have legs of lengths other than 5 and 12?

I'm really old, but I'll never be too old to become more educated.

- GmatMathPro
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Yes, there are infinitely many right triangles with a hypotenuse of 13 that have legs of lengths other than 5 and 12, but in this case the constraint of the perimeter forces it to be a 5-12-13 triangle.tomada wrote:We can assume that the other two sides are 50 and 120 when we're told that the diagonal is 130? I see that the dimensions satisfy the perimeter that we're given but, in general, could a triangle with a hypotenuse of 13 have legs of lengths other than 5 and 12?