Question

Two poles of height a meters and b meters are P meters apart. Prove that the height h of the point of intersection N of the lines joining the top of each pole to the foot of the opposite pole is $\frac{ab}{a+b}$ meters.

Answer 1

In $\Delta SBA$ and $\Delta NBC$ (Similar triangles)

$\frac{h}{b}=\frac{y}{x+y}\Rightarrow x+y=\frac{by}{h}\to \left(1\right)$

In $\Delta DAB$ and $\Delta NAC$ (Similar triangles)

$\frac{h}{a}=\frac{x}{x+y}\Rightarrow x+y=\frac{ax}{h}\to \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$

$\Rightarrow \frac{by}{h}=\frac{ax}{h}\Rightarrow \frac{b}{a}=\frac{x}{y}$

In equation $1$

$x+y=\frac{by}{h}$

Divide by $y$

$\Rightarrow \frac{x}{y}+1=\frac{b}{h}\Rightarrow \frac{b}{a}+1=\frac{b}{h}\Rightarrow \frac{h}{b}=\frac{a}{a+b}\Rightarrow h=\frac{ab}{a+b}$