In $\Delta PQR$,$Z$,$X$,and $Y$ are the midpoints of $\overline{PQ}$,$\overline{QR}$,and $\overline{PR}$ respectively. Prove that the correspondence $PQR \leftrightarrow XYZ$ is a similarity.

(A) $1$. Given: In $\Delta PQR$,$Z$,$X$,and $Y$ are midpoints of sides $\overline{PQ}$,$\overline{QR}$,and $\overline{PR}$ respectively.
$2$. By the Midpoint Theorem,the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of its length.
$3$. Therefore,$ZY \parallel QR$ and $ZY = \frac{1}{2} QR = XQ = XC$.
$4$. Similarly,$ZX \parallel PR$ and $ZX = \frac{1}{2} PR = YR = PY$.
$5$. Also,$YX \parallel PQ$ and $YX = \frac{1}{2} PQ = ZP = ZQ$.
$6$. In $\Delta PQR$ and $\Delta XYZ$,we have the ratios of corresponding sides:
$\frac{PQ}{YX} = \frac{QR}{XZ} = \frac{PR}{ZY} = 2$.
$7$. Since the ratios of all three pairs of corresponding sides are equal,by the $SSS$ (Side-Side-Side) similarity criterion,$\Delta PQR \sim \Delta YXZ$.
$8$. Note: The correspondence $PQR \leftrightarrow XYZ$ implies $\Delta PQR \sim \Delta YXZ$ based on the side ratios.