Find the perimeter of the shaded sector \( POR \).

The perimeter of the sector includes the arc \( PQR \) and the radii \( OP \) and \( OR \).

Given: Arc length \( PQR = 10 \, \text{cm} \), radius \( OP = OR = 4 \, \text{cm} \).

Perimeter = Arc \( PQR + OP + OR \):

\[ 10 + 4 + 4 = 18 \, \text{cm} \]

Answer: The perimeter of the shaded sector is \( 18 \, \text{cm} \).

Find \( \theta \).

Use the arc length formula: \( \text{Arc length} = r \theta \).

Given: Arc \( PQR = 10 \, \text{cm} \), radius \( r = 4 \, \text{cm} \).

\[ 10 = 4 \theta \]

\[ \theta = \frac{10}{4} = 2.5 \, \text{radians} \]

Answer: \( \theta = 2.5 \, \text{radians} \)

Find the area of the shaded sector \( POR \).

Use the sector area formula: \( \text{Area} = \frac{1}{2} r^2 \theta \).

Given: \( r = 4 \, \text{cm} \), \( \theta = 2.5 \, \text{radians} \) (from part b).

\[ \text{Area} = \frac{1}{2} \cdot 4^2 \cdot 2.5 = \frac{1}{2} \cdot 16 \cdot 2.5 = 8 \cdot 2.5 = 20 \, \text{cm}^2 \]

Answer: The area of the shaded sector is \( 20 \, \text{cm}^2 \).