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Math Help - Height of intersection POINT

  1. #1
    Junior Member mathbuoy's Avatar
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    Question Height of intersection POINT

    Two towers with heights a and b meters respectively are 100 meters apart.

    What is the height of the intersection point of the line joining the top of each tower to the base of the other?

    # show working and explanation PLS
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  2. #2
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    Hello, mathbuoy!

    Did you make a sketch?


    Two towers with heights a and b meters are 100 meters apart.

    What is the height of the intersection point of the lines
    joining the top of each tower to the base of the other?
    Code:
        P *
          |*
          | *
          |  *
          |   *
          |    *
        a |     *     * R
          |      *T * |  
          |       *   |
          |     * |   | b
          |   *  h| * | 
          | *     |  *|
        Q * - - - * - * S
            100-x U x

    The towers are: . PQ = a\,\text{ and }\,RS = b

    The lines cross at T, . h = TU

    Let US = x\text{, then }QU = 100-x


    Since \Delta TUS \sim \Delta PQS\!:\;\;\frac{h}{x} = \frac{a}{100} \quad\Rightarrow\quad h \:=\:\frac{ax}{100} .[1]

    Since \Delta TUQ \sim \Delta RSQ\!:\;\;\frac{h}{100-x} = \frac{b}{100}\quad\Rightarrow\quad h \:=\:\frac{b(100-x)}{100} .[2]


    Equate [1] and [2]: . \frac{ax}{100} \:=\:\frac{b(100-x)}{100} \quad\Rightarrow\quad x \:=\:\frac{100b}{a+b}


    Substitute into [1]: . h \:=\:\frac{a}{100}\left(\frac{100b}{a+b}\right) \quad\Rightarrow\quad \boxed{h\;=\; \frac{ab}{a+b}}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    An observation . . .
    Divide top and bottom by ab\!:\;\;h \;=\;\frac{\dfrac{ab}{ab}}{\dfrac{a}{ab} + \dfrac{b}{ab}} \;=\;\frac{1}{\dfrac{1}{a} + \dfrac{1}{b}}


    The answer is one-half the harmonic mean of a\text{ and }b.

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