Let us consider D PQR and D STU as shown in Figure 1 below. These two triangles have an area of 6, but their perimeters are different. Let us consider D ABC and D XYZ as shown in Figure 2 below. These two triangles have a perimeter of 30, but their areas are different. Do triangles exist with the same area and the same perimeter? The answer to the preceding question is yes because congruent triangles have the same area and the same perimeter. Now the question is, do non-congruent triangles exist with the same area and perimeter?

Consider three triangles with sides of lengths, 5,12, 13 and and . All these triangles are mutually non-congruent. Simple calculations will show that all the three triangles have the same area and the same perimeter. Is the above example a unique case? Given any triangle can you find a non-congruent triangle that has the same area and the same perimeter as the given triangle? For a given particular triangle, how many mutually non-congruent triangles can you find that have same the area and the perimeter of the given triangle?