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Question
D and E are the mid-points of the sides AB and AC of ∆ABC and O is any point on side BC. O is joined to A. If P and Q are the mid-points of OB and OC respectively, then DEQP is ______.
Options
a square
a rectangle
a rhombus
a parallelogram
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Solution
D and E are the mid-points of the sides AB and AC of ∆ABC and O is any point on side BC. O is joined to A. If P and Q are the mid-points of OB and OC respectively, then DEQP is a parallelogram.
Explanation:
In ΔABC, D and E are the mid-points of sides AB and AC, respectively.
By mid-point theorem,
DE || BC ...(i)
DE = `1/2` BC
Then, DE = `1/2` [BP + PO + OQ + QC]
DE = `1/2` [2PO + 2OQ] ...[Since, P and Q are the mid-points of OB and OC respectively]
⇒ DE = PO + OQ
⇒ DE = PQ
Now, in ΔAOC, Q and E are the mid-points of OC and AC respectively.
∴ EQ || AO and EQ = `1/2` AO [By mid-point theorem] ...(iii)
Similarly, in ΔABO,
PD || AO and PD = `1/2` AO [By mid-point theorem] ...(iv)
From equations (iii) and (iv),
EQ || PD and EQ = PD
From equations (i) and (ii),
DE || BC (or DE || PQ) and DE = PQ
Hence, DEQP is a parallelogram.
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