- #1
Bashyboy
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Homework Statement
Suppose we have four isosceles triangles with the same area, which must some whole number less than 29. Denote the the lower base and upper base of the i-th triangle with ##y_i## and ##x_i##, respectively. Furthermore, suppose that the angle between the side of length ##y_i## and the adjacent sides is ##45^\circ##. What will the lengths ##x_i## and ##y_i## be?
Homework Equations
The Attempt at a Solution
Because the angle is ##45^\circ##, the leg of the triangle and the height of the isosceles trapezoid must be equal; that is, ##h = \frac{y_i - x_i}{2}##. Therefore, the area is given by
##\displaystyle A = \frac{y_i + x_i}{2} h##
##\displaystyle A = \frac{y_i + x_i}{2} \frac{y_i - x_i}{2}##
In general, the two factors will not be whole numbers; for instance, ##\displaystyle \frac{y_i + x_i}{2}## will result in a whole number only if ##y_i + x_i## is an even whole number, which can occur when ##x_i## and ##y_i## have the same parity.
This is where I am unsure of how to proceed. I was told that I should look at all the integers in ##\{1,2,3,...,29\}## which could be factored into four different ways, where each term in the factorization has opposite parity. This doesn't really make sense two me. Does ##\displaystyle A = \frac{y_i + x_i}{2} \frac{y_i - x_i}{2}## basically say that the area can be factored into two even integers, and therefore we would want to look for four factorizations with even terms?