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. 9.26). Show that ar (ABCD) = ar (PBQR).
[Hint: Join AC and PQ. Now compare ar (ACQ) and ar (APQ).]

Solution:

Let us join AC and PQ.

ΔACQ and ΔAQP are lying on the same base AQ and existing between the same parallels AQ and CP.

According to Theorem 9.2: Two triangles on the same base (or equal bases) and between the same parallels are equal in area.

Area (ΔACQ) = Area (ΔAPQ)

Subtracting ar (ΔABQ) on both sides.

ar (ΔACQ) - ar (ΔABQ) = ar (ΔAPQ) - ar (ΔABQ)

ar (ΔABC) = ar (ΔQBP) ... (1)

Since AC and PQ are diagonals of parallelograms ABCD and PBQR respectively,

Therefore, ar (ΔABC) = 1/2 ar (ABCD)... (2)

Similarly, ar (ΔQBP) = 1/2 ar (PBQR )... (3)

From Equations (1), (2), and (3), we obtain

1/2 ar (ABCD) = 1/2 ar (PBQR)

ar (ABCD) = ar (PBQR) proved.

☛ Check: NCERT Solutions Class 9 Maths Chapter 9

Video Solution:

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. 9.26). Show that ar (ABCD) = ar (PBQR). [Hint: Join AC and PQ. Now compare ar (ACQ) and ar (APQ).]

Maths NCERT Solutions Class 9 Chapter 9 Exercise 9.3 Question 9

Summary:

If 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, then ar (ABCD) = ar (PBQR).

☛ Related Questions:

• Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar (AOD) = ar (BOC).
• In Fig. 9.27, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that i) ar (ACB) = ar (ACF) ii) ar (AEDF) = ar (ABCDE)
• A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented.
• ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar (ADX) = ar (ACY).[Hint: Join CX.]