Question

Show that the line segment joining the mid point of a pair of opposite side of a parallelogram divide it into two part of equal area

Answer 1

This is very simple but tricky...
Since, I cannot draw the geometrical figure here, i will give u the directions,
draw a parallelogram, ABCD
mark the mid point of line AB let the point be named as P; mark the mid point of line DC let the point be named as Q.
Join P&Q
Now AP=PB=DQ=QC and AD=PQ=BC(as p is mid point of AB and Q is mid point of DC)-----------------1
area of ADQP = AD*DQ
area of PQCB = PQ*QC
AD=PQ (by 1)
DQ=QC(by 1)
therefore area of ADQP = PQ*QC
But area of PQCB is also PQ*QC
therefore area of ADQP=Area of PQCB
 
                            ********* Hence proved********

Answer 2

Let us consider ABCD be a parallelogram in which E and F are mid-points of AB and CD. Join EF.

To prove: ar (|| AEFD) = ar (|| EBCF)

Let us construct DG ⊥ AG and let DG = h where, h is the altitude on side AB.

Proof:

ar (|| ABCD) = AB × h

ar (|| AEFD) = AE × h

= ½ AB × h ….. (1) [Since, E is the mid-point of AB]

ar (|| EBCF) = EF × h

= ½ AB × h …… (2) [Since, E is the mid-point of AB]

From (1) and (2)

ar (|| ABFD) = ar (|| EBCF)

Hence proved.

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