Hello, magentarita!

Your forgot to give us the question . . .

Cabins $\displaystyle B$ and $\displaystyle G$ are located on the shore of a circular lake,

and cabin $\displaystyle L$ is located near the lake.

Point $\displaystyle D$ is a dock on the lake shore, collinear with cabins $\displaystyle B$ and $\displaystyle L$.

The road between cabins $\displaystyle G$ and $\displaystyle L$ is 8 miles long and is tangent to the lake.

The path between cabin $\displaystyle L$ and dock $\displaystyle D$ is 4 miles long. Code:

B * * *
o *
* \ *
* \ *
\ x
* \ *
* \ *
* \ *
\ D
* o
* * \ 4
* * \
* o * - - - - - - o L
G 8

The only questions we can answer are:

. . What are the distances $\displaystyle BD$ amd $\displaystyle BL$ ?

If that is indeed the question, we should know this theorem:

. . If a tangent and a secant are drawn to a circle from an external point,

. . the tangent is the mean proportional between the external segment

. . of the secant and the entire secant.

In this problem, we have: .$\displaystyle \frac{LD}{LG} \:=\:\frac{LG}{LB} \quad\Rightarrow\quad \frac{4}{8} \:=\:\frac{8}{x+4}$

Then: .$\displaystyle 4x + 16 \:=\:64 \quad\Rightarrow\quad 4x \:=\:48 \quad\Rightarrow\quad x \:=\:12$

Therefore: .$\displaystyle BD = 12,\;\;BL = 16$ miles.