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

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Accepted Answer

Step 1: $	riangle ABQ \sim 	riangle ACR$ (AA similarity)
$\frac{AB}{AC} = \frac{QB}{RC} = \frac{y}{z}$ ... (i)
Step 2: Similarly, $	riangle CBQ \sim 	riangle CAP$
$\frac{BC}{AC} = \frac{QB}{PA} = \frac{y}{x}$ ... (ii)
Step 3: $\frac{AB}{AC} + \frac{BC}{AC} = \frac{y}{z} + \frac{y}{x}$
Step 4: $\frac{AB + BC}{AC} = y\left(\frac{1}{z} + \frac{1}{x}ight)$
$\frac{AC}{AC} = y\left(\frac{1}{z} + \frac{1}{x}ight)$
Step 5: $	herefore \frac{1}{x} + \frac{1}{z} = \frac{1}{y}$

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