Expert: Scott A Wilson - 2/23/2010

Question
The perimeter of a rectangle is 34 ft., and the diagonal is 13 ft long. what are the length and width of the rectangle?

Answer
Half of 34 is 17, and that is the sum of the sides.
Let the sides be A and B.

We know that A + B = 17 and A²+B²=13².

From the 1st equation, we can say the B = 17 - A.
Putting this in the 2nd equation gives (17-B)² + B² = 169.

Multiplying it out gives 17² - 2*17*B + B² + B² = 169.  Now 17² = 289 and 2*17 = 34.
So we have 289 - 34B + 2B² = 169.
Reverseing the left side and subtracting 169 from both sides gives 2B² - 34B +120 = 0.

Division by 2 gives B² - 17B  + 60 = 0.   This factors into (B-5)(B-12) = 0.

Thus, we say B = 5 or B = 12.  Given that, A is the other, so he sides are of length 5 and 12.
The longer is usually the length (12) and the shorter is usually the width (5).