Question

In the above figure, seg PA, seg QB and RC are perpendicular to seg AC. From the information given in the figure, prove that: `1/x + 1/y = 1/z`

Answer 1

In ΔPAC and ΔQBC
∠PCA = ∠QCB
∠PAC = ∠QBC

ΔPAC congurent to ΔQBC
`(PA)/(QB) = (AC)/(BC)`
`x/y = (AB)/(BC)`
`y/x = (BC)/(AC)`

In ΔRCA and ΔQBA
∠RAC = ∠QAB
∠RCA = ∠QBA
ΔRCA is congruent to ΔQBA
`(RC)/(QB) = (AC)/(AB)`

`z/y= (AC)/(AB)`
`y/z= (AB)/(AC)`

adding both eq
`y/z + y/z = (BC + AC)/(AC) = 1` 

`y/z + y/z = 1`

multiplying both sides by y

`1/x + 1/z = 1/y`