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 yi and the adjacent sides is $45^\circ$. What will the area $A$ lengths $x_i$ and $y_i$ be?
Because the angle is $45^\circ$, the leg of the triangle and the height of the isosceles trapezoid must be equal; that is, $\displaystyle h= \frac{y_i−x_i}{2}$. Therefore, the area is given by
$\displaystyle A= \frac{y_i+x_i}{2} h$
$\displaystyle A = \frac{y_i+x_i}{2} \frac{y_i−x_i}{2}$
This is where I was not certain as to how to proceed.
I was advised to look at all of the integers between $1$ and $29$, find the one that could that had four factorizations, and then determine the dimensions from such information.
I found that this integer was $24$. So, for instance, the first isosceles trapezoid dimensions could be found as such:
$24=2⋅12$
Therefore, we can set up the equations
$\displaystyle \frac{y_1+x_1}{2}=12$
and
$\displaystyle \frac{y_1−x_1}{2}=2$
I don't quite understand this, however. Why can I write $\displaystyle 12⋅2=\frac{y_1+x_1}{2}⋅\frac{y_1−x_1}{2}$, and have this imply $\displaystyle \frac{y_1+x_1}{2}=12$ and $\displaystyle \frac{y_1−x_1}{2}=2$?
Also, the notion of parity is to somehow factor into this problem.
How can I see this things arise naturally? I feel as though not enough justification has been provided. If someone could go through the steps with me, showing me how one idea naturally leads to next, I would greatly appreciate it.
Once you get to $\displaystyle A = \frac{y_i+x_i}{2} \frac{y_i−x_i}{2}$ you are expected to argue that $\displaystyle \frac{y_i+x_i}{2} $ and $\displaystyle \frac{y_i−x_i}{2}$ must both be integers. If one is not an integer, it is a half integer and so is the other, so the product cannot be an integer. Then, as you say, you can look at the various factorizations of $24$. This is what you want to see how it arises naturally-it is quite common. Each factorization (or each factorization subject to some condition) corresponds to one of the $i$s in your list. That is why you can use $24=12 \cdot 2$ to give $\frac {y_1+x_1}2=12, \frac {y_1-x_1}2=2,y_1=14,x_1=10$-the other factorizations will give the other possibilities, like $24=6 \cdot 4$ gives $\frac {y_2+x_2}2=6, \frac {y_2-x_2}2=4,y_2=10,x_2=2$ and so on. Parity comes in because $x_i,y_i$ must be both even or both odd to make the factors integral.