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To calculate the perimeter of the given object, we need to consider the lengths of the sides and the arc segment.

1. **Calculate the lengths of the line segments:**

   The object consists of line segments \(AD\), \(DC\), and \(CA\), and an arc segment \(DC\). 

   - **Segment \(AD\):**
     \[
     A(-2, 1) \quad \text{and} \quad D(-10, 28.2)
     \]
     The distance formula is given by:
     \[
     \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
     \]
     So,
     \[
     AD = \sqrt{(-10 - (-2))^2 + (28.2 - 1)^2} = \sqrt{(-8)^2 + 27.2^2} = \sqrt{64 + 739.84} = \sqrt{803.84} \approx 28.3
     \]

   - **Segment \(DC\):**
     \[
     D(-10, 28.2) \quad \text{and} \quad C(6, 28.2)
     \]
     Since the y-coordinates are the same,
     \[
     DC = |x_2 - x_1| = |6 - (-10)| = 16
     \]

   - **Segment \(CA\):**
     \[
     C(6, 28.2) \quad \text{and} \quad A(-2, 1)
     \]
     Using the distance formula again,
     \[
     CA = \sqrt{(6 - (-2))^2 + (28.2 - 1)^2} = \sqrt{(8)^2 + 27.2^2} = \sqrt{64 + 739.84} = \sqrt{803.84} \approx 28.3
     \]

2. **Calculate the length of the arc segment \(DC\):**

   The arc length \(s\) can be found using the formula:
   \[
   s = r \theta
   \]
   where \(r\) is the radius and \(\theta\) is the central angle in radians.

   - The radius \(r = 10\) (since \(D\) and \(C\) are on the circle centered at \(B(-2, 18)\)).
   - The central angle is \(76^\circ\).

   Convert the angle from degrees to radians:
   \[
   \theta = 76^\circ \times \left(\frac{\pi}{180}\right) \approx 1.3265 \text{ radians}
   \]

   So,
   \[
   s = 10 \times 1.3265 \approx 13.3
   \]

3. **Add the lengths to find the perimeter:**

   \[
   \text{Perimeter} = AD + DC + CA + s = 28.3 + 16 + 28.3 + 13.3 \approx 85.9
   \]

Therefore, the perimeter of the object to the nearest tenth is **85.9 units**.

Would you like more details or have any questions? Here are some follow-up questions you might consider:

1. How is the distance formula derived?
2. What are the properties of a circle's radius and its central angle?
3. How do you convert degrees to radians?
4. Can you explain how to calculate arc length in detail?
5. What are some real-world applications of calculating perimeters?

**Tip:** When calculating distances and lengths in coordinate geometry, always ensure you use consistent units and verify your computations for accuracy.

Learn how to calculate the perimeter of an object using coordinate geometry with a step-by-step solution. This problem covers key concepts such as the distance formula, arc length calculation, and coordinate geometry applications. Suitable for high school students.

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