How are the two polygons similar? How are they dissimilar?
The rectangle has greater area and greater perimeter than the triangle. See solution.
Practice makes perfect
We are going to determine, without doing any calculations, which of the polygons that has the greater area and which that has the greater perimeter. We begin by comparing the areas.
Area
Let's draw the two polygons in a diagram.
Just by looking at the diagram we notice that the height h of the triangle has the same length as the length l of the rectangle and that the triangle's base b is the same width as the rectangle's width w.
Let's move the triangle to the right so that the two polygons overlap each other.
The entire triangle fits on top of the rectangle and we can see that some of the rectangle, half of it to be precise, is not covered. Therefore we can conclude that the rectangle has a larger area than the triangle.
Perimeter
To determine which one of the polygons that has the larger perimeter, we again use the diagram where we have moved the triangle to the right so that it sits on top of the rectangle. Let's name the vertices.
We need to determine which path is the shorter of A ⇒ B ⇒ C ⇒ A and A ⇒ B ⇒ C ⇒ D ⇒ A. Since the first two legs of each path is identical they must have the same length. Let's now compare the remaining legs of the two paths.
C ⇒ A and C ⇒ D ⇒ A
In both of these we begin at C and end up at A. The shortest distance between two points is a straight line. In the triangle we go along a straight line, C to A. In the rectangle we go C to D to A which is not a straight line. We therefore know that the rectangle's perimeter must be greater.
