Answer
$a.\quad 3$
$b.\quad n-3$
$c.\displaystyle \quad \frac{1}{2}n^{2}-\frac{3}{2}n$
$d.\quad 20$
Work Step by Step
a.
The polygon has six sides. From one vertex: three diagonals.
b.
It will have $n-3 $ diagonals, \\
as we can not join the vertex with itself and its two neighboring vertices to form a diagonal.
c.
$\displaystyle \frac{n}{2}(n-3)\qquad...$apply the Distributive Property.
$=\displaystyle \frac{n^{2}}{2}-\frac{3n}{2}$
$=\displaystyle \frac{1}{2}n^{2}-\frac{3}{2}n$
d.
Substitute $n=8$ into the diagonal formula
$D=\displaystyle \frac{8}{2}(8-3)\qquad...$apply the Distributive Property.
$D=4\cdot 5$
$D=20$.
The polygon has 20 diagonals
