Problem Solving

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

Let x and y equal the length and width of the rectangle.
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

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

Awesome solution, shankar.ashwin!!

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

Cheers,
Brent