ANSWER
The coordinates of the third vertices are \((6,6)\) or \((1,-2)\).
EXPLANATION
Let the hypotenuse of the right angle triangle have vertices \((6,-2)\) and \((1,6)\).
The third point is the intersection of the horizontal line that passes through (6,-2), that has equation \(y=-2\) and the vertical line that passes through \((1,6)\) which also has equation \(x=1\).
Hence one possible coordinates of the third vertex is \((1,-2)\).
The third vertex could also be the intersection of the vertical line that passes through (6,-2), that has equation \(x=6\) and the horizontal line that passes through \((1,6)\) which also has equation \(y=6\).
Hence another possible coordinates of the third vertex is \((6,6)\).
See diagram in attachment
