Video Transcript
Which of the following polygons is
convex?
We have five options here. So, let’s remind ourselves what it
means for a polygon, which is a two-dimensional shape with straight sides, to be
convex. In a convex polygon, all of the
interior angles must be less than 180 degrees. Another property is found by adding
in the diagonals to the shape, that is, joining the nonadjacent vertices with
straight lines. When we do, we find that all of the
diagonals of a convex polygon lie completely within inside the shape. And in fact, this means we can
instantly disregard option (A). If, for example, we join this
vertex to this vertex, we see that the diagonal lies outside of the shape.
We have a similar concern over
shape (C). If we join the two vertices shown,
the diagonal lies outside the shape. For shape (D), this diagonal lies
outside the shape. And for shape (E), we have another
diagonal that lies outside the polygon itself. And so, shapes (A), (C), (D), and
(E) all have at least one diagonal that doesn’t lie within the polygon. And so, they cannot be convex. So, by the process of elimination,
that must leave us with option (B).
But we’ll check in two ways. Firstly, we’ll double check that
each of the diagonals lies within the polygon. When we do, we get this sort of
star shape. And we see that every single
diagonal does indeed lie within the polygon itself. But we might also check each of the
individual angles. This angle is less than 180. This angle is less than 180, as are
the remaining three. And so, the polygon that’s convex
is (B).
