Expert: Richard J. Raridon - 3/4/2009

Question
Hi, I was given a set of math problems, but this was the only one I couldn't figure out. I tried the pythagorean theorem and 1/2bh, then tried to see what happened when I plugged in random values for x, but that didn't work out too well.

Here's the question: A kite frame is to be made from six pieces of wood. The four pieces that form its border have been cut to the lengths indicated in the figure. (in the figure, the top two border-lengths of the kite are both 5, and the two on the bottom are 12). In the figure, x is half of the horizontal crosspiece. (basically, the horizontal crosspiece is 2x, and the vertical one bisects it).

a.) Find a formula for the area of the kite in terms of x.

I know that half of the product of the diagonals is the area of a kite, but I can't figure out how to find the length of the vertical crosspiece in terms of x. Once I know how to get a formula, I can then find the dimensions of the maximum area of the kite.
--thanks

Answer
Since I can't see your figure, I'm assuming that the 5 and 12 pieces come together at a right angle.  That means the vertical piece is 13.  So the x piece forms parts of two right triangles with the upper triangle consisting of x, 5 and a piece y.  The lower triangle is composed of x, 12 and (13-y).  So now you have
x^2 +y^2 = 5^2 and x^2 +(13-y)^2 = 12^2, solve for x
If all you want is a formula for the area, it's (1/2)x(13)