The given figure shows parallelograms ABCD and APQR. Show that these parallelograms are equal in area.

We know that, Opposite sides of || gm are equal and parallel. ∴ AB || DC and AR || PQ. We know that, The area of triangle is half that of a parallelogram on the same base and between the same parallels. From figure, || gm ABCD and △ ABR lies on same base AB and between same parallel lines AB and DC. ∴ Area of △ ABR = Area of || gm ABCD ⇒ Area of || gm ABCD = 2 Area of △ ABR .......(1) We know that, Area of triangles on the same base and between the same parallels lines are equal. △ ABR and △ APR lie on same base AR and between same parallel lines AR and PQ. ∴ Area of △ ABR = Area of △ APR ........(2) From equations (1) and (2), we get : ⇒ Area of || gm ABCD = 2 Area of △ APR .........(3) Also, || gm APQR and △ APR lies on same base AR and between same parallel lines AR and PQ. ∴ Area of △ APR = Area of || gm APQR .......(4) Using value of area of △ APR from equation (4) in (3), we get : ⇒ Area of || gm ABCD = Area of || gm APQR ⇒ Area of || gm ABCD = Area of || gm APQR. Hence, proved that the parallelograms ABCD and APQR are equal in area.

In the given figure, AD // BE // CF. Prove that :
area (△ AEC) = area (△ DBF)

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In the given figure, ABCD is a parallelogram. BC is produced to point X. Prove that :
area (△ ABX) = area (quad.ACXD)

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ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.

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The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :
ar.(△ ABC) = 8 × ar.(△ QSB)

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Mathematics

The given figure shows parallelograms ABCD and APQR. Show that these parallelograms are equal in area.

Theorems on Area

Related Questions

In the given figure, AD // BE // CF. Prove that :
area (△ AEC) = area (△ DBF)

In the given figure, ABCD is a parallelogram. BC is produced to point X. Prove that :
area (△ ABX) = area (quad.ACXD)

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :
ar.(△ ABC) = 8 × ar.(△ QSB)

Mathematics

The given figure shows parallelograms ABCD and APQR. Show that these parallelograms are equal in area.

Theorems on Area

Related Questions

In the given figure, AD // BE // CF. Prove that :area (△ AEC) = area (△ DBF)

In the given figure, ABCD is a parallelogram. BC is produced to point X. Prove that :area (△ ABX) = area (quad.ACXD)

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm2; find the area of parallelogram.

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :ar.(△ ABC) = 8 × ar.(△ QSB)

In the given figure, AD // BE // CF. Prove that :
area (△ AEC) = area (△ DBF)

In the given figure, ABCD is a parallelogram. BC is produced to point X. Prove that :
area (△ ABX) = area (quad.ACXD)

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :
ar.(△ ABC) = 8 × ar.(△ QSB)