Question

$P$ is the mid-point of side $AB$ to a parallelogram $ABCD$. A line through $B$ parallel to $PD$ meets $DC$ at $Q$ and $AD$ produced at $R$. Then $BR$ is equal to ______.

• A. $BQ$
• B. $\frac{1}{2}BQ$
• C. $2BQ$
• D. None of these

Answer 1

The correct option is C $2BQ$

In $△ABR$, $P$ is the mid-point of $AB$ and $DP||BR$

$\therefore D$ is the mid-point of $AR$ .....(Converse of mid-point theorem).

Then, $AR=2AD$

$⟹$ $AR=2BC$ (Opposite sides of parallelogram are equal, i.e. $AD=BC$)

$⟹$ $AD=BC$.

Also, $D$ is the mid-point of $AR$ and $DQ||AB$ ....(Since, $ABCD$ is parallelogram)

Then, $Q$ is the mid-point of $DC$ ....(Converse of mid-point theorem)

$\therefore BR=2BQ$.

Hence, option $C$ is correct.