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 Quadrilateral Side Query (Posted on 2015-10-31)
PQRS is a quadrilateral such that :
∠SPQ = ∠PSR and ∠PQS = ∠QRS.

Find each of RS and SP in terms of PQ, QR, and QS.

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 No Solution Yet Submitted by K Sengupta No Rating

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 Solution Comment 1 of 1
`From triangle QRS let`
`   a = /RSQ, b = /QRS, c = /SQR`
`From triangle PQS let`
`   d = /PQS, e = /SPQ, f = /QSP`
`   a + b + c = 180 = d +    e    + f                    = b + (a + f) + f`
`       ==> c = 2*f`
`Let T be the point on side RS such that QT bisects /SQR.`
`                  QR*RSTherefore, RT = ---------  and                  QR + QS`
`           QT^2 = QR*QS*sqrt[1 - (RS/[QR + QS])^2]           (1)`
`Checking the angles we see that triangles RTQ and QPSare similar. Therefore,`
`    RT     QT     QR   ---- = ---- = ----                                      (2)    QP     SP     QS`
`Combining (1) and (2) gives`
`   RS = QP*(QR + QS)/QS  and  `
`   SP = sqrt[ QS*(QS^2 - QP^2)/QR ]`
`QED`

 Posted by Bractals on 2015-11-01 09:43:08

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