geometry help!!

[smart_pirate]

I need help setting up and equation for this problem. Two poles, 30 feet and 50 feet tall, are 40 feet apart and perpendicular to the groung. the poles are supported by wires attached from the rop of each pole to the bottom of the other, as in the picture provided. A coupling is placed a C where he two wires cross. How high above the ground is the coupling? ( The blue line indicates the height of the coupling: X feet)

[attachment=0:ifa0029m]geo pro.jpg[/attachment:ifa0029m]

 

[soroban]

Hello, smart_pirate!

Two poles, 30 feet and 50 feet tall, are 40 feet apart and perpendicular to the ground.
The poles are supported by wires attached from the rop of each pole to the bottom of the other.
A coupling is placed a C where he two wires cross. How high above the ground is the coupling?

    A *
      |*
      | *
      |  *
      |   *
      |    *
   50 |     *     * C
      |      *E * |
      |       *   |
      |     * |*  | 30
      |   *  h| * |
      | *     |  *|
      * - - - * - *
      B 40-x  F x D

\(\displaystyle \text{The two ladders are: }\:AB = 50,\;CD = 30\;\;\hdots\;\;\text{and }BD = 40\)

\(\displaystyle \text{The wires, }AD,\:BC\text{, cross at }E.\)

\(\displaystyle \text{The height of the crossing is: }\:h = EF\)

\(\displaystyle \text{Let }x = FD\text{, then }40-x = BF\)


\(\displaystyle \text{Since }\Delta EFD \sim \Delta ABD\!:\;\;\frac{x}{h} \:=\:\frac{40}{50} \quad\Rightarrow\quad x \:=\:\frac{4}{5}h\;\;[1]\)

\(\displaystyle \text{Since }\Delta EFB \sim \Delta CDB\!:\;\;\frac{40-x}{h} \:=\:\frac{40}{30} \quad\Rightarrow\quad x \:=\:40 - \frac{4}{3}h\;\;[2]\)


\(\displaystyle \text{Equate [1] and [2]: }\: \frac{4}{5}h \:=\:40-\frac{4}{3}h \quad\Rightarrow\quad h \:=\:\frac{75}{4} \:=\:18\tfrac{3}{4}\text{ ft}\)

 

[TchrWill]

]I need help setting up and equation for this problem. Two poles, 30 feet and 50 feet tall, are 40 feet apart and perpendicular to the groung. the poles are supported by wires attached from the rop of each pole to the bottom of the other, as in the picture provided. A coupling is placed a C where he two wires cross. How high above the ground is the coupling? ( The blue line indicates the height of the coupling: X feet)

[attachment=0]geo pro.jpg[/attachment

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.