Find a triangle, quadrilateral and pentagon with integer side lengths whose areas form a set of three consecutive positive integers. Make the areas as small as possible subject to these constraints. Report you answer by giving the areas of each figure with the side lengths of those figure?
We use a rectangle for our quadrilateral with base $= 7$ and height $= 1$ to get an area of $7$.
We can use a triangle with $b = 1$, $h =12$ and legs $= 12$ to get an area of $6$.
We use a pentagon with side length $= 1$ and height $= 2$ to get an area of $5$.
This would leave us with our consecutive integers of $5, 6, 7$.
Does the height for the pentagon matter in this case?
Can we make these any smaller? The problem seems too simple to solve am I missing something concerning the geometric properties of these shapes?