Question

In the following figure, PA, QB and RC are each perpendicular to AC. Prove that `1/x+1/z=1/y`

In the given figure, PA, QB and RC are perpendicular to AC. If PA = x units, QB = units and RC = z units, prove that `1/x+1/z=1/y`.

Answer 1

We have, PA ⊥ AC, QB ⊥ AC and RC ⊥ AC

Let, AB = a and BC = b

In ΔCQB and ΔCPA

∠QCB = ∠PCA                       ...[Common]

∠QBC = ∠PAC                       ...[Each 90°]

Then, ΔCQB ~ ΔCPA              ...[By AA similarity]

`therefore"QB"/"PA"="CB"/"CA"`             ...[Corresponding parts of similar Δ are proportional]

`rArry/x=b/(a+b)`             ...(i)

In ΔAQB and ΔARC

∠QAB = ∠RAC                ...[Common]

∠ABQ = ∠ACR               ...[Each 90°]

Then, ΔAQB ~ ΔARC           ...[By AA similarity]

`therefore"QB"/"RC"="AB"/"AC"`               ...[Corresponding parts of similar Δ are proportional]

`rArry/z=a/(a+b)`             ...(ii)

Adding equations (i) & (ii)

`y/x+y/z=b/(a+b)+a/(a+b)`

`rArry(1/x+1/z)=(b+a)/(a+b)`

`rArry(1/x+1/z)=1`

`rArr1/x+1/z=1/y`