D, E and F are respectively the mid-points of the sides BC, CA and AB of a
. Show that
(i) BDEF is a parallelogram. (ii)
(iii)
Solution (i)
In
, D is the mid-point of BC and E is the mid-point of AC.
Therefore, by mid-point theorem, we have
(A line joining the mid-points of the two sides of a triangle is parallel to the third side.)
(1)
Similarly, by mid-point theorem, we have
(2)
From (1) and (2), we can say that
BDEF is a parallelogram.
(A quadrilateral is a parallelogram if both the pairs of opposite sides are parallel.)
Solution(ii)
We have already proved above that BDEF is a parallelogram. In the same way, we can prove that FECD and AEDF are parallelograms using mid-point theorem.
(3)
(Diagonal of a parallelogram divides it into two triangles of equal areas.)
And,
(4)
(Diagonal of a parallelogram divides it into two triangles of equal areas.)
And,
(5)
(Diagonal of a parallelogram divides it into two triangles of equal areas.)
We have
Putting (3), (4) and (5) in the above equation, we get
Solution (iii)
(6)
But, we have
(Proved above) (7)
We also have
(8)
(Parallelograms on the same base and between the same parallels are equal in area.)
Putting (7) and (8) in equation (6), we get
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