A nice algebra / geometry problem with lines

My son is working his way through Art of Problem Solving’s Introduction to Algebra book and was stuck on one of the challenge problems in the chapter about lines (chapter 8). We talked through the problem this morning.

Here’s the problem and his solution. He was able to find one of the points that satisfied the conditions of the problem, but he was surprised to learn that there was a second point:

Having learned about the second solution, I asked him to try to find it:

After he found the second point I wanted to show him that there was a little more to this problem that you might think. Here’s how we can get to a solution by using a little algebra and the Pythagorean theorem:

Rather than crunching through all of this algebra, I plugged the equation into Wolfram Alpha. He was pretty surprised to see what the general solution looked like 🙂

So, a fun problem and an pretty challenging one, too. I was happy to see that you could extend the “surprising” fact that there were actually two solutions not just one to a new “surprising” fact that there were actually infinitely many!