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Thread: [SOLVED] what is the answer to this problem

  1. #1
    dearlybelovedx3
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    [SOLVED] what is the answer to this problem

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  2. #2
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    Hello, dearlybelovedx3!

    In right triangle ABC, BD is the altitude to hypotenuse AC.
    $\displaystyle AC = 20,\;BD = 6,\;AD < DC$

    (a) Find $\displaystyle AD$ and $\displaystyle DC.$

    (b) To the nearest tenth, find $\displaystyle AB.$
    Code:
                B
                *
               *|  *
              * |     *
             *  |6       *
            *   |           *
           *    |              *
        A * * * + * * * * * * * * * C
             x  D      20-x
          : - - - - 20  - - - - - :

    We have three similar right triangles: .$\displaystyle \Delta ABC \sim \Delta BDC \sim \Delta ADB$

    Let $\displaystyle AD\,=\,x$, then $\displaystyle DC \,=\,20-x$

    Then: .$\displaystyle \frac{x}{6} \:=\:\frac{6}{20-x}\quad\Rightarrow\quad x^2-20x+36\:=\:0$

    . . and we have: .$\displaystyle (x-2)(x-18)\:=\:\quad\Rightarrow\quad x \:=\:2,\:18$

    (a) Therefore: .$\displaystyle AD \,=\,2,\;DC\,=\,18$


    (b) In right triangle ADB, we have: .$\displaystyle AB^2 \:=\:2^2 + 6^2 \quad\Rightarrow\quad AB \:=\:\sqrt{40}\:\approx\:6.3$

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