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Question
In below Fig, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ = `1/4` AC. If PQ produced meets BC at R, prove that R is a mid-point of BC.
Solution
Join B and D, suppose AC and BD out at O.
Then OC = `1/2` AC
Now,
CQ = `1/4` AC
⇒ CQ = `1/2` `[1/2 AC ]`
= `1/2` × OC
In Δ DCO, P and Q are midpoints of DC and OC respectively
∴ PQ || PO
Also in Δ COB, Q is the midpoint of OC and QR || OB
∴ R is the midpoint of BC
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