Question

In the adjoining figure; $MP,LR$ and $NQ$ are perpendiculars to the straight line $MLN$. If $MP=x,LR=z$ and $NQ=y$, prove that $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$.

Answer 1

In$△NMP$ and $△NLR$,

$PM\parallel RL$

So, $△NMP\sim △NLR$

$\frac{z}{x}=\frac{LN}{MN}$ ----------(1)

In $△MNP$ and $△MLR$,

$QN\parallel RL$

So, $△MNP\sim △MLR$

$\frac{z}{y}=\frac{ML}{MN}$ -----------(2)

Adding equation (1) and (2) we get,

$\frac{z}{x}+\frac{z}{y}=\frac{LN}{MN}+\frac{ML}{MN}$

or, $\frac{z}{x}+\frac{z}{y}=\frac{MN}{MN}$

or, $\frac{z}{x}+\frac{z}{y}=1$

or, $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$