Find that length

Define $i$ and $j$ to be the unit vectors in the horizontal and vertical directions respectively. Also let length $BD =a$ and length $BE = b$ .

$\vec{FG} = \vec{FE} + \vec{EC} + \vec{CG}$ $\vec{FG} = \frac{1}{2}\vec{DE} + \vec{EC} + \frac{1}{2}\vec{CA}$ $\vec{FG} = \frac{1}{2}(-ai + bj) + (3j) + \frac{1}{2}\bigg((a+4)i - (b+3)j\bigg)$ $2\vec{FG} = -ai + bj + 6j + ai + 4i -bj - 3j$ $2\vec{FG} = 4i + 3j$ $\vec{FG} = 2i + \frac32j$ $|FG| = \sqrt{2^2 + \bigg(\frac32\bigg)^2}$ $|FG| = \frac52 = \boxed{2.5}$