Hello, natbat77!
On a level plot of ground there stand two vertical posts, one 6 ft tall, the other 10 ft tall.
From the top of each pole a rope is stretched to the foot of the other pole.
How far above ground is the point where the ropes cross? Code:
* C
* |
* |
* |
* |
* |
A * * | 10
| * P * |
| * |
6 | * : * |
| * :h * |
| * : * |
B * - - - * - - - - - - - * D
a Q b
The poles are: .$\displaystyle AB = 6,\;\;CD = 10$
The ropes $\displaystyle AD$ and $\displaystyle BC$ cross at $\displaystyle P$.
$\displaystyle PQ = h$ is perpendicular to $\displaystyle BD.$
Let $\displaystyle a = BQ,\;\;b = QD$
In similar right triangles $\displaystyle PQD\text{ and }ABD\!:\;\;\frac{h}{b} \:=\: \frac{6}{a+b} \quad\Rightarrow\quad b \:=\:\frac{ah}{6-h}$ .[1]
In similar right triangles $\displaystyle PQB \text{ and }CDB\!:\;\;\frac{h}{a} \:=\:\frac{10}{a+b} \quad\Rightarrow\quad b \:=\:\frac{10a - ah}{h}$ .[2]
Equate [1] and [2]: .$\displaystyle \frac{ah}{6-h} \:=\:\frac{10a-ah}{a} \quad\Rightarrow\quad ah^2 \:=\:60a - 6ah - 10ah + ah^2 $
. . $\displaystyle 16ah \:=\:60a \quad\Rightarrow\quad h \:=\:\frac{60a}{16a} \quad\Rightarrow\quad\boxed{ h \:=\:\frac{15}{4}\text{ ft}}$