1. Cabins B and G

Cabins B and G are located on the shore of a circular lake, and cabin L is located near the lake. Point D is a dockon the lake shore and is collinear with cabins B and L. The road between cabins G and L is 8 miles long and is tangent to the lake. The path between cabin L and dock D is 4 miles long.

2. 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.

3. ok

Originally Posted by Soroban
Hello, magentarita!

Your forgot to give us the question . . .

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.
Tell me, how do you make such beautiful geometric shapes with your keyboard?