The sides of a trapezoid are 5, 8, 11, and 13, and its diagonals are also integer numbers; what are they?
I don't like using computers for exact calculations, because of rounding errors. I decided to try 5 as the base and 8 as the parallel side. Using coordinates, the "base" was from P=(0,0) to Q=(5,0), and the "top" was from S=(a,b) to R=(a+8,b). PS could be 11, so a²+b²=11². QR should then be 13, so (a+3)²+b²=13². Subtracting these equalities gets 6a+9=48, so a=13/2, so then b=½√315.Now, just doing numbers shows that PR=
9 and QS=
17, so that's the answer.
PS. If that hadn't worked out, I'm not sure I would have tried the other possibilities! :-)
Solution, by hand
