Let $S$ be the set of points $(x,y)$ in the coordinate plane that satisfy the inequalities

Question

\begin{cases}

x &\ge 0 \\

y &\ge 0 \\

x + y \le 1

\end{cases}

Then the set $S$ is a polygon. The length of the longest side in $S$ is $a\sqrt{b}$ (expressed in simplest radical form). Find $a+b$.

Answer 1

Graphing the inequalities, we find that $S$ is the region of the coordinate plane bounded by the lines $x=0$, $y=0$, and $x+y=1$. Thus, $S$ is a right triangle with legs of length $1$ and hypotenuse $1$, so the answer is $1+1=\boxed{2}$. The only other step that should be done is to say we exclude the line $y+x=1.$ this is because a line does not have area.