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Questions
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`.
Solution
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`
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