Question

x and y are respectively the mid points of non parallel sides PS and qr of a trapezium pqrs prove that xy parallel pq and xy=1/2(pq+rs)
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Answer 1

Given :- X and Y are respectively the mid points of non parallel sides PS and QR of a trapezium PQRS. Prove that XY parallel PQ and XY = 1/2(PQ + RS) .

Solution :-

Construction: Join QY and produce it to meet RS produced at T.

Now, in ∆PQY and ∆STY we have,

• PY = YS (Y is Mid - Point of PS.)
• ∠PYQ = ∠TYS (Vertically opposite angles.)
• ∠QPY = ∠YST (Alternate angles.)

So,

• ∆PQY ≅ ∆STY (By ASA.)

Therefore,

• PQ = ST (By CPCT.)
• QY = YT (By CPCT.)

Now, in ∆QRT we have,

• X is the midpoint of QR.
• Y is the midpoint of QT

Therefore,

• XY ∥ RT and,
• XY = (1/2)RT (By Mid - Point theorem.)

Hence,

→ XY = (1/2)[RS + ST]

→ XY = (1/2)[RS + PQ]

→ XY = (1/2)[PQ + RS] .

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