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

Question

Smart Achievers · Accepted Answer

In $\frac{1}{x} + \frac{1}{z} = \frac{1}{y}$ CAP and $∆$ CBQ

$∠$ CAP = $∠$ CBQ = 90°

$∠$ PCA = $∠$ QCB (common angle)

So, $∆$ CAP $~ ∆$ CBQ (By AA similarly Rule)

Hence, $\frac{B Q}{A P} = \frac{B C}{A C} \Rightarrow \frac{y}{x} = \frac{B C}{A C}$ ...(i)

Now, in $∆$ ACR and $∆$ ABQ

$∠$ ACR = $∠$ ABQ = 90°

$∠$ QAB = $∠$ RAC (common angle)

So, $∆$ ACR $~ ∆$ ABQ (By AA similarity Rule)

Hence, $\frac{B Q}{C R} = \frac{A B}{A C} \Rightarrow \frac{y}{z} = \frac{A B}{A C}$ ...(ii)

On adding eqs. (i) and (ii), we get

$\frac{y}{x} + \frac{y}{z} = \frac{B C}{A C} + \frac{A B}{A C} \Rightarrow y (\frac{1}{x} + \frac{1}{z}) = \frac{B C + A B}{A C} = \frac{A C}{A C}$

$\Rightarrow y (\frac{1}{x} + \frac{1}{z}) = 1 \Rightarrow \frac{1}{x} + \frac{1}{z} = \frac{1}{y}$

Solution

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

Want Us to Email you About Special Offers & Updates?

Solution

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

Want Us to Email you About Special Offers & Updates?

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