Geometry

I've been working forever to figure out this problem!

If two flagpoles are 10m and 70m tall and are 100m apart, find the height of the point where a line from the top of the first to the bottom of the second intersects a line from the bottom of the first to the top of the second.

I've drawn a picture but I can't do it after that! Please help!

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asked by Spencer
  1. This one is hard to answer without showing you a picture but here goes. The line from the top of the 10m pole to the base of the 70m pole is the hypotenuse of a right triangle. Dropping a vertical line from the point of intersection forms a second triangle on the end that is similar to the original and the sides are proportional. Label the vertical line z and the horizontal line x. You can make a proportion now from the two triangles. 10/100 = z/x. In a similar way a proportion can be made with the 70m pole. The vertical would still be z but the horizontal could be y which will make the proportion 70/100 = z/y. Cross multiplying the proportions gives 10x=100z and 70y=100z. Since both 10x and 70y are equal to 100z they are equal to each other BUT y = 100-x. With substitution you get 10x=70(100-x). Solve this equation for x then substitute the answer in the very first equation to find z. z is the height you want.

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    posted by Dianne
  2. If two flagpoles are 10m and 70m tall and are 100m apart, find the height of the point where a line from the top of the first to the bottom of the second intersects a line from the bottom of the first to the top of the second.

    The historical basis of the problem:

    Two ladders of different lengths are leaning against two buildings with their bases against the opposite building.
    Given: The heights where the ladders touch the buildings, A and B.
    Find: The height of the point where they cross, X.

    Assume the following picture:
    I*
    I.....*....................................................*{
    I...........*.......................................*...... I
    I..................*.........................*..............I
    IA......................*............*.....................I
    I..............................*............................{B
    I........................*.....{.....*......................I
    I................*............ I X.........*...............{
    I........*.....................I...................*........{
    I*________________ I______________ * I
    (C - y) C Y

    1--Let A and B = the two height of the ladders against the buildings.
    2--Let X = the height of the ladder crossing.
    3--From the figure, A/C = X/Y or AY = CX.
    4--Similarly, B/C = X/(C - Y) or BY = BC - CX.
    5--Y = CX/A = (BC - CX)/B from which X = AB/(A+B).

    Note - X is actually one half the harmonic mean of the two dimensions A and B, the harmonic mean being 2AB/(A + B).

    Therefore, the height of the crossing is totally independant of the distance between the two buildings.

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    posted by tchrwill

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