Thread: Find number of different triangles possible?

The sides of a triangle are a, b and c where a, b and c are integers and . If c is given, show that the number of different triangles is or , according as c is even or odd. Also, show that the number of isoceles or equilateral triangle is or , according as is even or odd.

The important fact here is that in a triangle with sides , it must be true that and

The last two inequalities are obvious given that , so we just need to see how many ways we can choose and so that

Suppose is even.

Then if , then must equal .
if , then can equal or .
..
if , then b can equal .

(So far we have ways to choose and .

Doing the same thing when , we find that there are more ways to choose, giving a total of triangles.

The other cases of this problem are not much different.