Perimeter

Since the ratio is $1:3:5$ , the total is $1+3+5=9$ . The angles are

$\dfrac{1}{9}(180)=20^\circ,$ $\dfrac{3}{9}(180)=60^\circ$ and $\dfrac{5}{9}(180)=100^\circ$

the shortest side is opposite the smallest angle

By sine rule ,

$\dfrac{x}{\sin 60}=\dfrac{10}{\sin 20}$ $\implies$ $x=10\left(\dfrac{\sin 60}{\sin 20}\right)$

By sine rule again,

$\dfrac{y}{\sin 100}=\dfrac{10}{\sin 20}$ $\implies$ $y=10\left(\dfrac{\sin 100}{\sin 20}\right)$

Finally, the perimeter is

$P=10+10\left(\dfrac{\sin 60}{\sin 20}\right)+10\left(\dfrac{\sin 100}{\sin 20}\right)=10+25.32+28.79\approx64~m$