In the given figure PA, QB and RC are each perpendicular to AC. If $AP = x$, $BQ = y$ and $CR = z$, then prove that $\frac{1}{x} + \frac{1}{z} = \frac{1}{y}$.

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Question

In the given figure PA, QB and RC are each perpendicular to AC. If $AP = x$, $BQ = y$ and $CR = z$, then prove that $\frac{1}{x} + \frac{1}{z} = \frac{1}{y}$.

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Accepted Answer

Step 1: $	riangle PAC \sim 	riangle QBC$
Step 2: $	herefore \frac{x}{y} = \frac{AC}{BC}$ or $\frac{y}{x} = \frac{BC}{AC}$ ... (i)
Step 3: $	riangle RCA \sim 	riangle QBA$
$	herefore \frac{z}{y} = \frac{AC}{AB}$ or $\frac{y}{z} = \frac{AB}{AC}$ ... (ii)
Step 4: Adding (i) and (ii)
$\frac{y}{x} + \frac{y}{z} = \frac{BC + AB}{AC} = \frac{AC}{AC} = 1$
Step 5: $\Rightarrow \frac{1}{x} + \frac{1}{z} = \frac{1}{y}$

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