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Math Help - Triangles

  1. #1
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    Triangles

    Triangle ABC is isosceles with AC=BC=13 and AB=10. M and N are the feet of the perpendiculars from A and B to Bc and AC respectively. Find Bn.



    Could someone please help me? Thank you!
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  2. #2
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    Hello, Coach!

    Triangle ABC is isosceles with AC=BC=13,\;AB=10.
    M \text{ and }N are the feet of the perpendiculars from A \text{ and }B\text{ to }BC\text{ and }AC, resp.
    Find BN.
    Code:
                    C
                    *
                   /|\
                  / | \
                 /  |  \
                /   |   \
            13 /    |    \ 13
              /   12|     \
             /      |      \
            /       |       \
           /        |        \
          * - - - - + - - - - *
          A    5    D    5    B
    We see that AD = 5,\;AC = 13.
    . . From Pythagorus: . CD = 12




    Code:
                    C
                    *
                   /|\
                  / | \
                 /  |  \
                /   |   \
            N  /    |    \
              *     |     \
             /    * |      \
            /       | *     \
           /        |     *  \
          * - - - - + - - - - *
          A         D         B
    We have similar right triangles: . \Delta BNA \sim \Delta CDA
    . . (They share \angle A.)




    Code:
                                                C
                                                *
                                               /|
                                              / |
                                             /  |
                                            /   |
              N                        13  /    | 12
              *                           /     |
             /    *                      /      |
            /         *                 /       |
           /              *            /        |
          * - - - - - - - - - *       * - - - - *
          A        10         B       A    5    D
    Hence, we have: . \frac{BN}{10} \:=\:\frac{12}{13}


    Therefore: . \boxed{BN \:=\:\frac{120}{13}}

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