Parallelogram, Triangle Areas

In parallelogram ABCD, M and N are midpoints of sides AD and BC, respectively. DN intersects AB at P, CM intersects AB at Q, and DN intersects CM at O. If the area of parallelogram ABCD is 24 square cm, find the area of triangle QPO.


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Master plan: △QPO = ABNM + △MNO + △QAM + △BPN.

ABNM = ½ ABCD.

△MNO = ½△MNC (½ the base MC, same alititude).
△MNO = ½(½MNCD) = ½(½(½ABCD)) = 1/8 ABCD.

△QAM = △CDM = ½(MNCD) = ½(½ABCD) = ¼ABCD.
Likewise, △PBN = ¼ABCD.

So (see the top): △QPO = ½ABCD + 1/8 ABCD + ¼ABCD + ¼ABCD = 9/8 ABCD = 9/8 · 24 = 27 square cm.

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