If the mid points of the sides of a quadrilateral are joined in order, prove that the are of the parallelogram so formed will be half of the area of the given quadrilateral. Please experts answer it as quickly as possible.

given: ABCD is a quadrilateral. P, Q, R and S are the mid-points of the sides AB, BC , CD and AD respectively.

TPT: area (PQRS)=1/2 * area(ABCD).

construction: join AC and BD.

proof:

in the triangle ABD, P and S are the mid-points of the sides AB and AD respectively.

area(ΔASP)=1/4*area(ABD)

area(ΔASP)=1/4*1/2*area(ABCD)

area(ΔASP)=1/8*area(ABCD).........(1)

similarly

area(ΔBPQ)=1/8*area(ABCD).......(2)

area(ΔCQR)=1/8*area(ABCD)..........(3)

area(ΔRDS)=1/8*area(ABCD)........(4)

adding (1),(2),(3) and (4):

area(ΔASP)+area(ΔBPQ)+area(ΔCQR)+area(ΔRDS)=4*1/8*area(ABCD)=1/2*area(ABCD)......(5)

area(PSQR)=area(ABCD)-[area(ΔASP)+area(ΔBPQ)+area(ΔCQR)+area(ΔRDS)]

=area(ABCD)-1/2*area(ABCD)

=1/2area(ABCD)

hope this helps you.

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