Q: 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 Fig. ). Show that .
[Hint: Join AC and PQ. Now compare and .]
Join the AC and PQ.
It is given that ABCD is a ||gm and AC is a diagonal of ||gm
Therefore, ar(ABC) = ar(ADC) = 1/2 ar(||gm ABCD).............(i)
Also, ar(PQR) = ar(BPQ) = 1/2 ar(||gm PBQR).............(ii)
Since AQC and APQ are on the same base AQ and between same parallels AQ and CP.
ar(AQC) = ar (APQ)
Now, subtracting ABQ from both sides we get,
ar(AQC) - ar (ABQ) = ar (APQ) - ar (ABQ)
ar(ABC) = ar (BPQ)............(iii)
From eq(i), (ii) and (iii) we get