# Math Help - Height of intersection POINT

1. ## 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

2. 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 $a\text{ and }b.$