# Cabins B and G

• Oct 16th 2008, 06:20 PM
magentarita
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.
• Oct 16th 2008, 08:38 PM
Soroban
Hello, magentarita!

Your forgot to give us the question . . .

Quote:

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.

• Oct 17th 2008, 02:30 AM
magentarita
ok
Quote:

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?