Question

In triangle ABC, the medians BP and CQ are
produced upto points M and N respectively such
that BP = PM and CQ= QN. Prove that
(i) M, A and N are collinear.
(ii) A is the mid-point of MN.​

Answer 1

Answer:

Given: BP and CQ are medians of AB and AC respectively of triangle ABC

BP and CQ are produced to M and N such that BP = PM and CQ = QN

In △APM and △BPC,

AP=PC

PM=BP

∠APM=∠BPC                       ...(Vertically opposite angles)

therefore, △APM≅△BPC              ...(SAS rule)

∠AMP=∠PBC                    ...(By cpct)

Similarly, △AQN≅△BPC 

hence, ∠ANQ=∠QBC        ..(By cpct)

Hence, N, A, M lie on a straight line.

NM=NA+AM=BC+BC=2BC

hence, A is the mid point of MN