In the adjoining figure, show that ABCD is a parallelogram. Calculate

1.	In the adjoining figure, show that ABCD is a parallelogram. Calculate the area of parallelogram ABCD.
Answer» Construct AM ⊥ DC and CL ⊥ AB. Extend the lines DC and AB and join AC. We know that Area of the quadrilateral ABCD = area of triangle ABD + area of triangle DCB From the figure we know that Area of △ ABD = Area of △ DCB So it can be written as Area of the quadrilateral ABCD = 2 (Area of △ ABD) We get Area of △ ABD = ½ (area of quadrilateral ABCD) ……… (1) Area of quadrilateral ABCD = area of △ ABC + area of △ CDA From the figure we know that Area of △ ABC = Area of △ CDA So it can be written as Area of the quadrilateral ABCD = 2 (Area of △ ABC) We get Area of △ ABC = ½ (area of quadrilateral ABCD) ……… (2) Using the equations (1) and (2) Area of △ ABD = Area of △ ABC = ½ BD = ½ CL It can be written as CL = BD In the same way we get DC = AB and AD = BC Hence, ABCD is a parallelogram Area of parallelogram ABCD = base × height = 5 × 7 = 35 cm² Therefore, area of parallelogram ABCD is 35 cm².

In the adjoining figure, show that ABCD is a parallelogram. Calculate the area of parallelogram ABCD.

Answer»

Construct AM ⊥ DC and CL ⊥ AB. Extend the lines DC and AB and join AC.

We know that

Area of the quadrilateral ABCD = area of triangle ABD + area of triangle DCB

From the figure we know that

Area of △ ABD = Area of △ DCB

So it can be written as

Area of the quadrilateral ABCD = 2 (Area of △ ABD)

We get

Area of △ ABD = ½ (area of quadrilateral ABCD) ……… (1)

Area of quadrilateral ABCD = area of △ ABC + area of △ CDA

From the figure we know that

Area of △ ABC = Area of △ CDA

So it can be written as

Area of the quadrilateral ABCD = 2 (Area of △ ABC)

We get

Area of △ ABC = ½ (area of quadrilateral ABCD) ……… (2)

Using the equations (1) and (2)

Area of △ ABD = Area of △ ABC = ½ BD = ½ CL

It can be written as

CL = BD

In the same way we get

DC = AB and AD = BC

Hence, ABCD is a parallelogram

Area of parallelogram ABCD = base × height

= 5 × 7

= 35 cm²

Therefore, area of parallelogram ABCD is 35 cm².