Question

The given figure shows the parallelograms ABCD and APQR.
Show that these parallelograms are equal in the area.
[ Join B and R ]

Answer 1

Join B and R and P and R.
We know that the area of the parallelogram is equal to twice the area of the triangle if the triangle and the parallelogram are on the same base and between the parallels.
Consider ABCD parallelogram:

Since the parallelogram ABCD and the triangle ABR lie on AB and between the parallels AB and DC, we have
Area(`square`ABCD ) = 2 x Area( ΔABR )        ....(1)

We know that the area of triangles with the same base and between the same parallel lines are equal.
Since the triangles ABR and APR lie on the same base AR and between the parallels AR and QP, we have,
Area ( ΔABR ) = Area ( ΔAPR )               ....(2)

From equations (1) and (2), we have,
Area(`square`ABCD) = 2 x Area( ΔAPR )        .....(3)

Also, the triangle APR and the parallelograms, AR and QR, lie on the same base AR and between the parallels, AR and QP,
Area( ΔAPR ) = `1/2` x Area(`square`ARQP )    ....(4)

Using (4) in equation (3), We have,
Area(`square`ABCD ) = 2 x `1/2 xx "Area"( square`ARQP )

Area( `square"ABCD" ) = "Area"( square` ARQP)
Hence Proved.