Question

Triangle PQR is an isosceles triangle. The side QP is produced to S. Such tha PQ=PS,prove tha angle QRS=90 degree.

Answer 1

You have not said, but this only works if P is the vertex between the equal sides.

So, let the base be 2b and each of the two equal sides be s.

Then the altitude is h and

b^2+h^2 = s^2

That is, if the midpoint of the base is M, then

QM^2 + PM^2 = PQ^2

Now, if PQ=PS, S is the midpoint of QP and M is the midpoint of QR

That means that ∆QMP ~ ∆QRS and since ∆QMP is a right triangle, so is ∆QRS, making R a right angle.