$ABCD$ is a quadrilateral in which $P$,$Q$,$R$ and $S$ are mid-points of the sides $AB$,$BC$,$CD$ and $DA$ respectively (see figure). $AC$ is a diagonal. Show that: $PQ = SR$.

(N/A) Given: $P, Q, R, S$ are mid-points of sides $AB, BC, CD, DA$ respectively of quadrilateral $ABCD$. $AC$ is a diagonal.
To prove: $PQ = SR$.
Proof:
In $\triangle ABC$,$P$ is the mid-point of $AB$ and $Q$ is the mid-point of $BC$.
By the Mid-point Theorem,the line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.
Therefore,$PQ \parallel AC$ and $PQ = \frac{1}{2} AC$ ........ $(1)$
In $\triangle ADC$,$S$ is the mid-point of $AD$ and $R$ is the mid-point of $CD$.
By the Mid-point Theorem,$SR \parallel AC$ and $SR = \frac{1}{2} AC$ ........ $(2)$
From equations $(1)$ and $(2)$,since both $PQ$ and $SR$ are equal to $\frac{1}{2} AC$,we get $PQ = SR$.