The Distance Length

Let $E$ be the origin of the $xy$ -plane. The equations of $CE$ , $DE$ , and $BF$ are as follows:

$\begin{cases} CE: & y = \dfrac 65 x \\ DE: & y = - \dfrac 65 x \\ BF: & y = \dfrac 3{10}x - \dfrac 32 \end{cases}$

Then the coordinates of $G(x_G, y_G)$ and $H(x_H, y_H)$ are:

$\begin{cases} \dfrac 65 x_G = \dfrac 3{10}x_G - \dfrac 32 & \implies x_G = - \dfrac 53 & \implies y_G = \dfrac 65 \left(-\dfrac 53\right) = -2 \\ - \dfrac 65 x_H = \dfrac 3{10}x_H - \dfrac 32 & \implies x_H = 1 & \implies y_H = - \dfrac 65 \left(1\right) = - \dfrac 65 \end{cases}$

Then we have:

$\begin{aligned} GH & = \sqrt{\left(-\frac 53 - 1\right)^2 + \left(-2+\frac 65\right)^2} \\ & = \sqrt{\frac {64}9 + \frac {16}{25}} = \boxed{\dfrac 4{15}\sqrt{109}} \end{aligned}$

Hope you understand, my solution is in the figure below.