P1= side 9 inches
P2= side 15 inches
what’s is the ratio of their areas?
the area of a pentagon is:
[tex]\text{area}=\frac{\text{perimeter}\cdot\text{apothem}}{2}[/tex]so first we have to find the perimeters
[tex]\begin{gathered} P1=9\cdot5=45 \\ P2=15\cdot5=75 \end{gathered}[/tex]and now we have to find the apothem
[tex]\begin{gathered} x1 \\ 4.5^2+x1^2=9^2 \\ x1^2=81-20.25 \\ x1=\sqrt[]{60.75} \\ x1=7.79 \end{gathered}[/tex][tex]\begin{gathered} x2 \\ 7.5^2+x2^2=15^2 \\ x2^2=225-56.25 \\ x2=\sqrt[]{168.75} \\ x2=12.99 \end{gathered}[/tex]finally, we can find the area
[tex]\begin{gathered} P1 \\ A1=\frac{45\cdot7.79}{2}=175.27 \\ P2 \\ A2=\frac{75\cdot12.99}{2}=487.12 \end{gathered}[/tex]the ratio of their areas are:
[tex]\frac{A1}{A2}=\frac{175.27}{487.12}=0.36\approx\frac{9}{25}[/tex]