Question

AB, CD, PQ are pendicular to BD. If AB = x, CD = y and PQ = z, prove that

1/x+1/y+1/z​

Answer 1

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Given,

AB ,CD , PQ are perpendicular to BD ,

AB = x , CD = y and PQ = z

To prove:-

${1/x+1/y+1/z}$

Proof:-

Consider ∆ABD and ∆PQD

∠ABD = ∠PQD = 90°

∠ADB = ∠PDQ ( common angle )

By A.A similarity ,

∆ABD ≅ ∆PQD

PQ/AB = DQ/BD

➪z/x = QD/BD ( c.p.s.t ) ---------------( 1 )

Consider ∆CDB and ∆PQB

∠CDB = ∠PQB = 90°

∠CBD = ∠PBQ ( common angle )

∆CDB ~ ∆PQB ( A.A similarity )

So , z/y = BQ/BD ------------------------( 2 )

From ( 1 ) and ( 2 ) we get

z/x + z/y = QD/BD + BQ/BD

➪z( 1/x + 1/y ) = ( QD + BQ )/BD

➪z( 1/x + 1/y ) = BD/BD

➪z( 1/x + 1/y ) = 1

➪1/x + 1/y = 1/z

Hence,proved..............................

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