Math's: Quadrilaterals: Relationships of Quadrilaterals with Triangles

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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 .

Parallelograms and its Relation

Parallelograms and Its Relation

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

Triangles Between the Same Parallels

Triangles between the Same Parallels

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 Between the Same Parallels

Triangles between the Same Parallels

  • 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 .

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