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## The Diagonal of a Parallelogram and Its Relation to the Area

• A diagonal of a parallelogram divides it into two triangles of equal area. This is explained with the help of an illustration as follows-

• Draw a parallelogram . Join its diagonal . and .

Consider the two triangles and in which the parallelogram has been divided by the diagonal . Because , therefore .

Now, Area of ....(i)

Area of ...(ii)

As, and

∴ Area of Area of

## Parallelograms and Triangles between the Same Parallels

Two parallelograms or triangles, having same or equal bases and having their other vertices on a line parallel to their bases, are said to be on the same or equal bases and between the same parallels. This can be proved as follows

Given: Parallelograms and & stand on the same base and between the same parallels and

To prove: (i) Area of

(ii) Area of Area of

We have, (Opposite sides of a parallelogram)

and (Opposite sides of a parallelogram)

...(i)

Now, ...(ii)

...(iii)

From (i), (ii) and (iii), we get

Join the diagonals and of the two parallelograms and respectively. We know that a diagonal of a divides it in two triangles of equal area.

[Each half of ]

and [Each half of ]

[Since area of = Area of ]