# Investigation

Find triangles having integer area and integer sides.

What approach?

Is there an infinite number of them?

Are there common characteristics for those you have found?

Try using Heron's Formula for the area of a triangle

Specific Problems:

A. Find a triangle with perimeter 12 having integer area and integer sides.

B. Find a triangle having integer sides and integer area that is not a right triangle. Can you find others? Generalize.

C. Find the smallest perimeter for which there are two different triangles with integer sides and integer area.

D. Find 5 triangles with perimeter of 100 units having integer area and integer sides.

Four of them are isosceles. One is scalene.

E. Find all the triangles with perimeter of 84 having integer sides and integer area.

How many? Does your approach assure you that you have found all of them?

• See Perfect Triangles -- Perfect triangles are triangles with integer area and integer sides AND the area and perimeter are numerically equal.