In the given figure, DDD is the midpoint of side ABABAB of ΔABCΔABCΔABC and PPP is any point on BCBCBC. If CQ||PDCQ||PDCQ||PD meets ABABAB in QQQ, prove that ar(ΔBPQ)ar(ΔBPQ)ar(ΔBPQ) is equal to 12ar(ΔABC)12ar(ΔABC)12ar(ΔABC).
Answer:
- We are given that DDD is the midpoint of side ABABAB of ΔABCΔABCΔABC and PPP is any point on BCBCBC.
Also, CQ||PDCQ||PDCQ||PD meets ABABAB in QQQ. - Let us join CDCDCD and PQPQPQ.
- We know that a median of a triangle divides it into two triangles of equal area.
In ΔABCΔABCΔABC, CDCDCD is a median. [Math Processing Error] - But, ΔDPCΔDPC and ΔDPQΔDPQ being on the same base DPDP and between the same parallels DPDP and CQCQ, we have: [Math Processing Error] Using (i)(i) and (ii)(ii), we get: [Math Processing Error]