If the mid points of the sides of a quadrilateral are joined in order, prove that the are of the - Maths - Areas of Parallelograms and Triangles

Question

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

Ajanta Trivedi · Accepted Answer

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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(Î”ASP)=1/4*area(ABD)
area(Î”ASP)=1/4*1/2*area(ABCD)
area(Î”ASP)=1/8*area(ABCD) (1)
similarly
area(Î”BPQ)=1/8*area(ABCD) (2)
area(Î”CQR)=1/8*area(ABCD) (3)
area(Î”RDS)=1/8*area(ABCD) (4)
adding (1),(2),(3) and (4):

area(Î”ASP)+area(Î”BPQ)+area(Î”CQR)+area(Î”RDS)=4*1/8*area(ABCD)=1/2*area(ABCD)
(5)

area(PSQR)=area(ABCD)-[area(Î”ASP)+area(Î”BPQ)+area(Î”CQR)+area(Î”RDS)]
=area(ABCD)-1/2*area(ABCD)
=1/2area(ABCD)
hope this helps you

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.