The following term: $$\sqrt{2017^2-2018^2+2019^2}$$ is the same as this term: $$\sqrt{2018^2+2}$$ how can one show without the need of a calculator that these are the same?
The original question was the following:
How long is the distance between point A and D - the diagonals are orthogonal and the other distances as labeled
I then called the point where the diagonals meet $M$ and with the help of the Pythagoras' theorem I came up with the following equations:
$$ AM^2 + BM^2 = 2017^2 $$ $$ AM^2 + CM^2 = 2018^2 $$ $$ CM^2 + DM^2 = 2019^2 $$ $$ DM^2 + AM^2 = AD^2 $$
By rearrange the first three equations I had the following term:
$$AD^2 = 2017^2-2018^2+2019^2 $$ as the solution given was: $$ AD = \sqrt{2018^2+2}$$ I was curious how to tell it was the same
Are there any other "nicer" ways to find this solution?
the root comes from the actual problem - but it's not important
It's still important in the sense that theactual problem
might well have a simpler or more direct solution without going through those square roots. See What is the XY problem?. $\endgroup$