Compute the perimeter of the triangle determined by $$ A(-2,-3), B(2,-6) $$, and $$ C(5,-5) $$. $$ A=\square $$ unit $$ ^{2} $$

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To compute the perimeter of the triangle determined by the points $$ A(-2,-3) $$, $$ B(2,-6) $$, and $$ C(5,-5) $$, we need to find the lengths of the sides of the triangle formed by these points.

1. **Calculate the distance between points $$ A $$ and $$ B $$**:
   The distance $$ d $$ between two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by the formula:
   $$
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   $$
   For points $$ A(-2, -3) $$ and $$ B(2, -6) $$:
   $$
   d_{AB} = \sqrt{(2 - (-2))^2 + (-6 - (-3))^2} = \sqrt{(2 + 2)^2 + (-6 + 3)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
   $$

2. **Calculate the distance between points $$ B $$ and $$ C $$**:
   For points $$ B(2, -6) $$ and $$ C(5, -5) $$:
   $$
   d_{BC} = \sqrt{(5 - 2)^2 + (-5 - (-6))^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10}
   $$

3. **Calculate the distance between points $$ C $$ and $$ A $$**:
   For points $$ C(5, -5) $$ and $$ A(-2, -3) $$:
   $$
   d_{CA} = \sqrt{(-2 - 5)^2 + (-3 - (-5))^2} = \sqrt{(-7)^2 + (2)^2} = \sqrt{49 + 4} = \sqrt{53}
   $$

4. **Calculate the perimeter of the triangle**:
   The perimeter $$ P $$ of the triangle is the sum of the lengths of its sides:
   $$
   P = d_{AB} + d_{BC} + d_{CA} = 5 + \sqrt{10} + \sqrt{53}
   $$

Now, let's compute the numerical value of the perimeter.

I'll calculate $$ \sqrt{10} $$ and $$ \sqrt{53} $$ to find the total perimeter.

Calculating these values:
- $$ \sqrt{10} \approx 3.162 $$
- $$ \sqrt{53} \approx 7.280 $$

Now, summing these values:
$$
P \approx 5 + 3.162 + 7.280 \approx 15.442
$$

Thus, the perimeter of the triangle is approximately $$ 15.442 $$ units.

The final answer is:
$$
P \approx 15.442 \text{ units}
$$
The perimeter of the triangle is approximately 15.442 units.

Compute the perimeter of the triangle determined by \( A(-2,-3), B(2,-6) \), and \( C(5,-5) \). \( A=\square \) unit \( ^{2} \)

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