Answer
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 $\triangle FMN=11+11+4=26$.
