Two line segments $AB$ and $AC$ include an angle of $60^{\circ}$ where $AB = 5 \, cm$ and $AC = 7 \, cm$. Locate points $P$ and $Q$ on $AB$ and $AC$,respectively,such that $AP = \frac{3}{4} AB$ and $AQ = \frac{1}{4} AC$. Join $P$ and $Q$ and measure the length $PQ$.

(D) Given: $AB = 5 \, cm$ and $AC = 7 \, cm$.
Also,$AP = \frac{3}{4} AB$ and $AQ = \frac{1}{4} AC$.
Calculation of lengths:
$AP = \frac{3}{4} \times 5 = 3.75 \, cm$.
$AQ = \frac{1}{4} \times 7 = 1.75 \, cm$.
Construction Steps:
$1$. Draw $AB = 5 \, cm$.
$2$. Draw a ray $AZ$ making an angle of $60^{\circ}$ with $AB$.
$3$. Mark point $C$ on $AZ$ such that $AC = 7 \, cm$.
$4$. To locate $P$ on $AB$ such that $AP = \frac{3}{4} AB$,divide $AB$ in the ratio $3:1$ using the standard division of a line segment method.
$5$. To locate $Q$ on $AC$ such that $AQ = \frac{1}{4} AC$,divide $AC$ in the ratio $1:3$ using the standard division of a line segment method.
$6$. Join $PQ$.
$7$. By measurement,the length $PQ \approx 3.25 \, cm$.