Question

State whether following statement is true or false.

$P$ is the midpoint of the sides $CD$ of a parallelogram $ABCD$. A line through $C$ parallel to $PA$ intersect $AB$ at $Q$ and $DA$ produced at $R$ then $DA=AR$ and $CQ=QR$.

• A. True
• B. False

Answer 1

The correct option is A True

$BC=AD$ [ Opposite sides of a parallelogram ]

$BC\parallel AD$ [ Opposite sides of a parallelogram ]

$DC=AB$ and $DC\parallel AB$ [ Opposite sides of a parallelogram ]

Now, it is given that $P$ is mid-point of $CD$

$DP=PC=\frac{1}{2}DC$

Now,

$QC\parallel AP$ and $PC\parallel AQ$

As the opposite sides are equal and parallel.

$\therefore$ $APCQ$ is a parallelogram.

$AQ=PC=\frac{1}{2}DC=\frac{1}{2}AB=BQ$

Now,

In $△AQR$ and $△BQC,$

$\Rightarrow$ $AQ=BQ$

$\Rightarrow$ $\angle AQR=\angle BQC$ [ Vertically opposite angles ]

$\Rightarrow$ $\angle ARQ=\angle BCQ$ [ Alternate angles ]

$\therefore$ $△AQR\cong △BQC$ [ By $ASA$ theorem ]

$\Rightarrow$ $AR=BC$ [ By C.P.C.T ]

$\Rightarrow$ $BC=DA$

$\Rightarrow$ $AR=DA$

$\Rightarrow$ $CQ=QR$