Question

In figure, $\square ABCD$ is a parallelogram, $P$ and $Q$ are mid-points of side $AB$ and $DC$ respectively, then prove $\square APCQ$ is a parallelogram.

Answer 1

Hint: Since, $P$ and $Q$ are mid-points of side $AB$ and $DC$ respectively, this implies $AP = \dfrac{1}{2}AB$ and $CQ = \dfrac{1}{2}CD$. We will use the given condition, to find the relation between opposite sides of $\square APCQ$. If opposite sides are equal and parallel, then the quadrilateral is a parallelogram.

Complete step-by-step answer:
Here, we are given that $\square ABCD$ is a parallelogram.
In a parallelogram, opposite sides are parallel and equal.
Therefore, we can say $AB\parallel CD$ and $AB = CD$
As $P$ and $Q$ are mid-points of side $AB$ and $DC$ respectively.
$AP = \dfrac{1}{2}AB$ and $CQ = \dfrac{1}{2}CD$
Therefore, $CQ = \dfrac{1}{2}AB$
Hence, $AP = CQ$
Also, $AP$ is a part of $AB$ and $CQ$ is a part of $CD$.
Then, we can say $AP\parallel CQ$
Therefore, if opposite sides are equal and parallel, then the quadrilateral $\square APCQ$ is a parallelogram.

Note: A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral, where opposite sides are equal and parallel. In a parallelogram opposite angles are equal. When the perpendicular distance between two lines is the same at the corresponding points, then the lines are parallel.