Diagonals of a rhombus are equal and perpendicular to each other. Is this statement true? Give reason for your answer.

$(1)$ The quadrilateral obtained by joining in order the midpoints of the sides of a rhombus is a .......
$(2)$ The quadrilateral obtained by joining in order the midpoints of the sides of a rectangle is a ..........

Can all the angles of a quadrilateral be right angles? Give reason for your answer.

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a rhombus. If $AB = 12 \, \text{cm}$, then the perimeter of $ABCD$ is $48 \, \text{cm}$.
$(2)$ $PQRS$ is a square. If $PR = 8 \, \text{cm}$, then the perimeter of $PQRS$ is $32 \, \text{cm}$.

In $\Delta ABC$,the points $M$ and $N$ are on sides $AB$ and $AC$ respectively such that $AM = \frac{1}{4} AB$ and $AN = \frac{1}{4} AC$. Prove that $MN = \frac{1}{4} BC$.

In the figure,$P$ is the mid-point of side $BC$ of a parallelogram $ABCD$ such that $\angle BAP = \angle DAP.$ Prove that $AD = 2 CD.$