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The given problem shows an equilateral triangle \( OPQ \) with a side length of 12 cm, and it has a shaded region formed by a circular arc \( PXQ \) with center \( O \) and radius 12 cm. Additionally, the point \( S \) is such that \( OS = \frac{1}{3} \times 12 = 4 \) cm. You are asked to find the **perimeter of the shaded region**.

### Steps to Solve the Problem:

1. **Length of Arc \( PXQ \):**
   The arc \( PXQ \) subtends the angle at the center \( O \), which is 60 degrees because the triangle \( OPQ \) is equilateral. To calculate the length of this arc, we use the formula for arc length:
   \[
   \text{Arc length} = \theta \cdot r
   \]
   where \( \theta \) is the angle in radians, and \( r \) is the radius of the circle. The angle 60° in radians is:
   \[
   \theta = \frac{\pi}{3} \, \text{radians}
   \]
   Therefore, the arc length becomes:
   \[
   \text{Arc length} = \frac{\pi}{3} \times 12 = 4\pi \, \text{cm}
   \]

2. **Lengths of Straight Lines \( OP \) and \( OQ \):**
   Since the triangle is equilateral, the straight lines \( OP \) and \( OQ \) both have a length equal to the radius, which is 12 cm.

3. **Perimeter of the Shaded Region:**
   The perimeter consists of the arc length \( PXQ \) and the two straight lines \( OP \) and \( OQ \). Thus, the total perimeter \( P \) is:
   \[
   P = OP + OQ + \text{Arc length} = 12 + 12 + 4\pi
   \]
   Simplifying:
   \[
   P = 24 + 4\pi \, \text{cm}
   \]

So, the **perimeter of the shaded region** is \( 24 + 4\pi \) cm.

Would you like further details on any part of the solution, or do you have other questions?

### Related Questions:
1. What is the area of the shaded region in terms of \( \pi \)?
2. How would the solution change if the triangle was not equilateral but isosceles?
3. Can you derive the area of sector \( OPXQ \) formed by the arc?
4. What is the total circumference of the circle from which the arc is derived?
5. How would you calculate the length of arc if the angle was 45 degrees instead of 60?

**Tip:** Always convert angles to radians when working with circular arcs and sectors, as it simplifies the application of formulas!

The diagram shows an equilateral triangle OPQ of side 12 cm, and the point S such that OS = 1/3 × 12 = 4 cm. The arc PXQ has center O and radius 12 cm. Find the perimeter of the shaded region.

This problem involves finding the perimeter of a shaded region formed by a circular arc in an equilateral triangle with a side length of 12 cm. The solution uses the arc length formula and involves angle conversion. Suitable for Grades 9-10 students.

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