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).
A C B Q D P


Answer:


Step by Step Explanation:
  1. 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.
  2. Let us join CDCDCD and PQPQPQ.
    A C B Q D P
  3. 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]
  4. 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]

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