Question

In $△$ ABC, the medians $BP$ and $CQ$ are produced upto points M and N respectively such that $BP=PM$ and $CQ=QN$. Hence, A is the mid-point of MN.
If the above statement is true then mention answer as 1, else mention 0 if false.

Answer 1

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 $△BPC$,
$AP=PC$
$PM=BP$
$\angle APM=\angle BPC$ ...(Vertically opposite angles)
therefore, $△APM\cong △BPC$ ...(SAS rule)
$\angle AMP=\angle PBC$ ...(By cpct)
Similarly, $△AQN\cong △BPC$
hence, $\angle ANQ=\angle QBC$ ..(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