In given figure the side $A B$ of a parallelogram $A B C D$ is produced to any point $P$ . A line through A and parallel to $C P$ meets $C B$ produced at $Q$ and then parallelogram $P B Q R$ is completed (see Fig). Show that:
$a r e a (A B C D) = a r e a (P B Q R)$

Open in App

Solution

Join $A C$ and $P Q$ .
Since $A C$ and $P Q$ are diagonals of parallelogram $A B C D$ and parallelogram $B P Q R$ respectively.
Therefore, Area $(A B C) = \frac{1}{2} A r e a (A B C D)$ ...... (1)
and Area $(P B Q) = \frac{1}{2}$ Area $(B P R Q)$ ... (2)

Triangles $A C Q$ and $A Q P$ are on the same base $A Q$ and between the same parallels $A Q$ and $C P$ .
$a r (A C Q) = a r (A Q P)$
$\Rightarrow a r (A C Q) - a r (A B Q) = a r (A Q P) - a r (A B Q)$ (Subtracting $a r (A B Q)$ from both sides)
$\Rightarrow a r (A B C) = a r (B P Q)$

$\frac{1}{2} a r (A B C D) = \frac{1}{2} a r (B P R Q)$ (Using (1) and (2))
$\Rightarrow a r (A B C D) = a r (B P R Q)$

Suggest Corrections

Similar questions

Q. The side

A B

of a parallelogram

A B C D

is produced to any point

P

. A line through

A

and parallel to

C P

meets

C B

produced at

Q

and then parallelogram

P B Q R

is completed (see Fig.) Show that

a r (A B C D) = a r (P B Q R)

Q. The side AB of a parallelogram ABCD is produced to any point P.
A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed (see the following figure).
Show that ar (ABCD)=ar(PBQR).

The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed (see the following figure). Find the relation between area (ABCD) and area (PBQR). [3 MARKS]

Q. Question 9
The side AB of a parallelogram ABCD is produced to any point P.
A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed. (see the following figure).
Show that ar(ABCD) = ar(PBQR).

Q. The side AB of parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed. Show that ar(ABCD)=ar(PBQR).

Join BYJU'S Learning Program

In given figure the side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed (see Fig). Show that:area(ABCD)=area(PBQR)

In given figure the side $A B$ of a parallelogram $A B C D$ is produced to any point $P$ . A line through A and parallel to $C P$ meets $C B$ produced at $Q$ and then parallelogram $P B Q R$ is completed (see Fig). Show that:
$a r e a (A B C D) = a r e a (P B Q R)$