Hello, and welcome to FMH!

Normally we don't generally provide complete solutions, but since you are a mom helping your son, I'm going to do so.

First we need to figure out the length of the unknown side. The Pythagorean theorem tells us we may state:

\(\displaystyle x^2+5^2=13^2\)

\(\displaystyle x^2=13^2-5^2=169-25=144=12^2\)

Now, since \(x\) represent a length of a line, it cannot be negative, so we find:

\(\displaystyle x=12\)

If one endpoint of the side 12 units long is at (1,3), and this side lies along a vertical line, then the other endpoint could be at:

\(\displaystyle (1,3\pm12)=(1,3(1\pm4))\)

Notice this represent two possible points...one above and one below.

And so the mid-point of that side would be:

\(\displaystyle (x,y)=\left(\frac{1+1}{2},\frac{3+3(1\pm4)}{2}\right)=\left(1,3(1\pm2)\right)\)

Does that make sense?