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Through A, B and C lines RQ, PR and QP h...

Through A, B and C lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and AB of a `Delta`ABC as shown in figure. Show that `BC=(1)/(2)QR`.

Text Solution

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Given : Tringles ABC and PQR in which `AB "||"QP,BC"||"RQ and CA"||"PR.`
To prove : `BC=1/2QR`
`"Proof: Since" RA"||"BC and BR"||"CA" "("given")`
`therefore"Quadrilateral RBCA is a parallelogram."" "("pair of opposite sides are parallel")`
`therefore" "RA=BC" "(because"opposite sides of a parallelogram")...(1)`
Now, quadrilateral ABQA is a parallelogram.
`therefore" "AQ=BC" "(because"opposite sides of a parallelogram")...(2)`
Adding (1) and (2), we get
`" "RA+AQ=2BC`
`implies" "QR=2BC`
`implies" "BC=1/2QR`
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