Perimeters

We can easily eliminate the first two options.

1. Let us assume that a triangle with a smaller perimeter will always have a larger area. Then let us take an equilateral triangle with side length 1, and another, with say a side length of 100. It can easily be seen that the former triangle will have a smaller perimeter, but also a smaller area. This example eliminates the first option.

2. The second option states that a triangle with a greater perimeter will always have a greater area. Then, let us take a simple example, consisting of two triangles, $\triangle ABC$ and $\triangle PQR$ , such that

   • The sides of $\triangle ABC$ are 3, 4 and 6 units in length; $perimeter = 3 + 4 + 6 = 13 units$
   • The sides of $\triangle PQR$ are all 4 units in length; $perimeter = 4 + 4 + 4 = 12 units$

   The perimeter of $\triangle ABC$ is greater than the perimeter of $\triangle PQR$ ; if option 2 is true, then, the area of $\triangle ABC$ must be greater than the area of $\triangle PQR$ . Calculating the areas using Heron's Formula, we get:

   • $ar(\triangle ABC) \approx 5.33$
   • $ar(\triangle PQR) \approx 6.93$

   Clearly, the area of $\triangle ABC$ is lesser than the area of $\triangle PQR$ , which eliminates option 2.

This means that there is no absolute relation between the perimeter of two triangles and their areas. Therefore, the last option is true.