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in Pythagoras Theorem by (47.6k points)
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From the information given in the adjoining figure, prove that PM = PN = √3 × a

1 Answer

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Best answer

Proof: 

In ∆PMR,

QM = QR = a [Given] 

∴ Q is the midpoint of side MR. 

∴ seg PQ is the median. 

∴ PM2 + PR2 = 2PQ2 + 2QM2 [Apollonius theorem]

∴ PM2 + a2 = 2a2 + 2a2 

∴ PM2 + a2 = 4a2 

∴ PM2 = 3a2 

∴ PM, = (√3a) (i) [Taking square root of both sides] 

SimlarIy, in ∆PNQ, R is the midpoint of side QN. 

∴ seg PR is the median. 

∴ PN2 + PQ2 = 2 PR2 + 2 RN2 [Apollonius theorem]

∴ PN2 + a2 = 2a2 + 2a2 

∴ PN2 + a2 = 4a2 

∴ PN = 3a2 

∴ PN = (√3a) (ii) [Taking square root of both sides] 

∴ PM = PN = √3a [From (i) and (ii)]

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