Answer
The perimeter of $\triangle FGH$ is $37$.
Work Step by Step
In order to find the perimeter of $\triangle FGH$, we need to find the lengths of all its sides. We are already given the lengths of two sides, so we just need the length of the third side, $FG$.
$\overline{IJ}$ joins the midpoint of $\overline{FH}$ and also the midpoint of $\overline{GH}$. According to the triangle midsegment theorem, if a line segment joins two sides of a triangle at their midpoints, then that line segment is parallel to the third side of that triangle and is half as long as that third side.
Knowing this information, we can deduce that $\overline{FG}$, which is the third side, is parallel to $\overline{IJ}$, which is the line segment that joins the midpoints of the other two sides of the triangle. It also means that $\overline{FG}$ is two times the length of $\overline{IJ}$. So, if we know $IJ$, then we can find $FG$:
$FG = 2(IJ)$
Let's plug in the values we know:
$FG = 2(7)$
Multiply to solve for $FG$:
$FG = 14$
Now, we can find the perimeter of the triangle, which is the sum of all the sides of the triangle:
$P = FH + GH + FG$
Let's plug in the values for each of the sides.
$P = 10 + 13 + 14$
Add from left to right:
$P = 37$
