# Math՚s: Quadrilaterals: Relationships of Quadrilaterals with Triangles (For CBSE, ICSE, IAS, NET, NRA 2022)

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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 ]

- Triangles on the same or equal bases having equal areas have their corresponding altitudes equal.
- We know that
- Thus, if it is given that
- and . Then,
- Altitudes of and are equal in length. That is .