The perimeter of $\triangle FMN$ is 26.
Work Step by Step
Given $\triangle FJH$ is an isosceles triangle and Given $M$ and $N$ are midpoints, then $\triangle FMN$ is also an isosceles triangle. Since $N$ is the midpoint of $FH$, and $NH=5y-9$, then $FN=5y-9$. Since $\triangle FMN$ is an isosceles triangle, then $FM=FN$. $FM=FN$ $2y+3=5y-9$ Subtract $2y$ from both sides... $3=3y-9$ Add nine to both sides... $12=3y$ Divide by 3... $4=y$ If $y=4$ and $\triangle FMN$ is an isosceles triangle with $FM=FN$, then $FM=FN=11$. Also, the mid-segment of an isosceles triangle is half the length of the base. The base is $2y$ or $8$ in length. Thus the mid-segment is $4$ in length. Therefore the perimeter of $\triangle FMN=11+11+4=26$.