Question

Find the total number of the triangles whose all the sides are integer and longest side is of 10 in length. If
the similar clause is applied for the isosceles triangle then what will be the total number of triangles?

Answer 1

As we know, the sum of two sides of triangle must be bigger than the length of the greatest side.

We have possible pairs of sides for triangle:

$(6,5)(7,4)(7,5)(7,6)(7,8)(7,9)(8,3)(8,4)(8,5)(8,6)(8,7)(8,9)(9,2)(9,3)(9,4)(9,5)(9,6)(9,7)(9,8)$.

There are 19 pairs ( (one side, other side) of triangle which is not isosceles). As for the isosceles triangles there will be only 4 pairs: $(6,6)(7,7)(8,8)(9,9)$.