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  1. #1
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    Question

    As shown in the diagram, a cottage is located at C on an island in the lake, and a marine is located at B. If the distance from A to D is 10.0 km, and <ABC = <CAB = 28˚, find the distance from B to C.

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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, Ph4m!

    As shown in the diagram, a cottage is located at C on an island in the lake,
    and a marine is located at B.
    If the distance AD = 10.0 km, and <ABC = <CAB = 28, find distance BC.
    Code:
                                  A
                                  *
                               * **
                            *   * *
                         * 28 *  *
                      *       *   * 10.0 km
                   *         *x   *
                *           *     *
             * 28    124 * 56  *
          * * * * * * * * * * * * *
          B       x       C       D
    Let $\displaystyle x = BC$.

    Since $\displaystyle \angle ABC = \angle CAB = 28^o$, triangle ABC is isosceles.
    Hence: .$\displaystyle AC = BC = x$.
    Since $\displaystyle \angle ACB \:= \:180^o - 28^o - 28^o \:=\:124^o$, then: .$\displaystyle \angle ACD = 56^o$

    In right triangle ADC, we have: .$\displaystyle \sin56^o \:=\:\frac{10}{x}\quad\Rightarrow\quad x \:=\:\frac{10}{\sin56^o} \:=\:12.06217949$

    Therefore: .$\displaystyle BC \:\approx\:12.1$ km.

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