Answer
$DF = 24$
Work Step by Step
$\overline{BE}$ joins the midpoint of $\overline{AD}$ and also the midpoint of $\overline{DF}$. 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 $\overline{DF}$, which is the third side, is parallel to $\overline{BE}$, which is the line segment that joins the midpoints of the other two sides of the triangle. It also means that $\overline{DF}$ is two times the length of $\overline{BE}$. So, if we know $BE$, then we can find $DF$:
$DF = 2(BE)$
Let's plug in the values we know. Once we find what $x$ is, we can find the value of $DF$:
$5x + 9 = 2(2x + 6)$
Use the distributive property on the right side of the equation:
$5x + 9 = 4x + 12$
Subtract $4x$ from each side of the equation to isolate the $x$ term on the left side of the equation:
$x + 9 = 12$
Subtract $9$ from each side of the equation to solve for $x$:
$x = 3$
Now, we can find $DF$:
$DF = 5x + 9$
Plug in $3$ for $x$:
$DF = 5(3) + 9$
Multiply first, according to order of operations:
$DF = 15 + 9$
Add to solve for $DF$:
$DF = 24$
