A number of years ago I was asked by a colleague to help with a young lad’s maths homework. It took me a couple of hours to work out the length of the plank, using the method described here. Since the problem was given to a class of 3rd year high school pupils, and I needed to use a method I learned in sixth year, I was a little suspicious. Apparently the only hint given was that the method used similar triangles, which I had done. However, if you assume that the small triangles between the ends of the plank and the corners of the room are 0.3m:0.4m:0.5m then the large triangles are of sides 7.7m and 5.6m, so that the length of the plank is 9.521m by Pythagoras. This is actually 3mm short, but a more troubling consequence is that 32 equals 18.
The flaw in the argument? Well, the sides of the plank aren’t parallel with the diagonal of the room. The small triangles aren’t 0.3m:0.4m:0.5m at all.