Question

PLEASE HELP

Answer 1

In the smallest of the three triangles (with sides $a,b,3$) we have

$a^2+3^2=b^2$

In the second-largest triangle (with sides $a,108$), the length of the missing side is $\sqrt{a^2+108^2}$. Then in the largest triangle (with sides $b,111,\sqrt{a^2+108^2}$), we have

$b^2+\left(\sqrt{a^2+108^2}\right)^2=(108+3)^2=111^2$

$\implies b^2+a^2+108^2=111^2$

$\implies b^2+a^2=657$

Equivalently, we have

$(a^2+9)+a^2=657\implies 2a^2=648\implies a^2=324\implies a=18$

In turn, we find

$b^2=18^2+3^2=298\implies b=\sqrt{333}=3\sqrt{37}$