A Geometry Word Problem: Crossing Ladders

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matthew042

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There are two flagpoles, one of height 12 and one of height 16. A rope is connected from the
top of each flagpole to the bottom of the other. The ropes intersect at a point EF units above
the ground. Find EF. In the attachment is a diagram of the problem. I need the answer and step by step instructions on how to get there.
 
Re: A Geometry Word Problem

Clearly by corresponding angles we have \(\displaystyle \vartriangle AEB \approx \vartriangle DEC\).
Because \(\displaystyle \overleftrightarrow {AB}\parallel \overleftrightarrow {CD}\).
 
Re: A Geometry Word Problem

Yes, what about the two triangles that have the same angles?
 
Re: A Geometry Word Problem

Hello, matthew042!

There are two flagpoles, one of height 12 and one of height 16.
A rope is connected from the top of each flagpole to the bottom of the other.
The ropes intersect at a point EF units above the ground. .Find EF.
Click to expand...
Code: 
                              * C
                            * |
                          *   |
                        *     |
                      *       |
                    *         |
    A *           *           | 16
      |   *   E *             |
      |       *               |
   12 |     * :   *           |
      |   *   :h      *       |
      | *     :           *   | 
    B *-------+---------------* D
      : - x - F - - - y - - - :

\(\displaystyle \text{Let }\,h = EF,\:x = BF,\:y = FD.\)

\(\displaystyle \text{Since }\Delta ABD \sim \Delta EFD\!:\;\frac{h}{y} \,=\,\frac{12}{x+y} \quad\Rightarrow\quad h \:=\:\frac{12y}{x+y}\) .[1]

\(\displaystyle \text{Since }\Delta CDB \sim \Delta EFB\!:\;\frac{h}{x} \,=\,\frac{16}{c+y} \quad\Rightarrow\quad h \:=\:\frac{16x}{x+y}\) .[2]

\(\displaystyle \text{Equate [1] and [2]: }\;\frac{16x}{x+y} \:=\:\frac{12y}{x+y} \quad\Rightarrow\quad y \:=\:\frac{4}{3}x\)

\(\displaystyle \text{Substitute into [2]: }\;h \;=\;\frac{16x}{x+\frac{4}{3}x} \;=\;\frac{16x}{\frac{7}{3}x} \quad\Rightarrow\quad \boxed{h\;=\;\frac{48}{7}}\)

 
Re: A Geometry Word Problem

There are two flagpoles, one of height 12 and one of height 16. A rope is connected from the
top of each flagpole to the bottom of the other. The ropes intersect at a point EF units above
the ground. Find EF. In the attachment is a diagram of the problem. I need the answer and step by step instructions on how to get there.

The distance EF is actually one half the harmonic mean of the two pole heights, the harmonic mean being 2AB/(A + B). Therefore, Therefore, the height of the crossing is totally independant of the distance between the two buildings.


The more famous version

Given: The lengths of the two ladders (25 and 30 ft.) and the height of the crossing point of the two ladders (10 ft.).
FInd: The distance between the two buildings, c.

The Ultimate Challenge

Two ladders are leaning against two adjacent buildings such that each ladder rests at the base of one building and leans against the opposite building, crossing each other somewhere in between. Let the bases of the buildings be A on the left and E on the right. One ladder extends from E to point B on the opposite building and the other extends from A to point F on the opposite building, crossing one another at point C. Point D lies directly below point C in between the buildings.

Determine the smallest set of values for the lengths AB, AC, AD, AE, AF, BC, BE, CD, and EF such that they are all integers..
The lengths of the two ladders are not equal.

Assume the following picture:

B
I*
I * F
I * * I
I * * I
I * C * I
I * I
I * I * I
I * I * I
I * I * I
I*____________________________ I________________________ * I
A..............................D...........................E
 
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