# Determining third vertex of the right angled isosceles triangle

If A(9,-9), B(1,-3) are the vertices of a right angled isosceles triangle, then the third vertex is?? Here in this question i got stuck to the point that which side is taken as the given coordinates. Whether it will be equal sides or hypotenuse??? If we consider both then how many will be the solution???

• ya that is the main issue...I think upto this instant, question is incomplete... – Pratyush Dec 29 '14 at 10:17
• Either the question is incomplete or we will have to make many cases. – Dheeraj Kumar Dec 29 '14 at 10:17
• exactly, i just want to know the various cases – Pratyush Dec 29 '14 at 10:18
• @Pratyush Consider that the side $\overline{AB}$ is the hypotenuse in which case the third vertex $C$ can occur at 2 different points. If it is one of the legs, then the third vertex can occur at 4 different points. – Chinmay Nirkhe Dec 29 '14 at 10:20

1. AB as hypotenuse $\rightarrow$ two positions for the third vertex.
2. AB as an equal side $\rightarrow$ now there will be four positions for the third vertex. i.e. Taking either A or B as the $90^{\circ}$ angle.