I don't believe this is possible, geoNewbie
The hypotenuse will be the fixed distance from O to A = sqrt (10^2 + 10^2) = sqrt (200) =OA
Call the point that we seek ( x, x+ 6)
The distance from O to Q = sqrt [ ( x^2 + ( x + 6)^2 ] = OQ
And the distance from A to Q = sqrt [ ( x - 10) + ( x + 6 - - 10)^2 ] =
sqrt [ (x - 10)^2 + (x + 16)^2 ] = AQ
Using the Law of Cosines we have that
OA^2 = OQ^2 + AQ^2 - 2( OQ * AQ) cos (90)
The cos 90 = 0 so this boils down to
OA^2 = OQ^2 + AQ^2
200 = ( x^2 + (x + 6)^2 + ( x - 10)^2 + ( x + 16)^2
Using WolframAlpha to solve this, we get no real solutions for x
Also.....playing around with this triangle in Geogebra, I couldn't find any angles of OQA even approaching 90°
In fact about the largest angle I could find is ≈ 38.xxx °

I wonder if you might not mean angle QOA is the right angle ????
