To find the perimeter of the triangle with vertices $A(-7, -5)$, $B(-3, -5)$, and $C(-7, -2)$, we need to calculate the lengths of all three sides of the triangle and then sum them up.

Step 1: Calculate the lengths of the sides

1. Length of side $AB$: $AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$ Substituting the coordinates: $AB = \sqrt{(-3 - (-7))^2 + (-5 - (-5))^2} = \sqrt{(-3 + 7)^2 + (0)^2} = \sqrt{4^2 + 0} = \sqrt{16} = 4$

2. Length of side $BC$: $BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2}$ Substituting the coordinates: $BC = \sqrt{(-7 - (-3))^2 + (-2 - (-5))^2} = \sqrt{(-7 + 3)^2 + (-2 + 5)^2} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

3. Length of side $CA$: $CA = \sqrt{(x_A - x_C)^2 + (y_A - y_C)^2}$ Substituting the coordinates: $CA = \sqrt{(-7 - (-7))^2 + (-5 - (-2))^2} = \sqrt{(0)^2 + (-5 + 2)^2} = \sqrt{0 + (-3)^2} = \sqrt{9} = 3$

Step 2: Calculate the perimeter

The perimeter $P$ is the sum of the lengths of the sides: $P = AB + BC + CA = 4 + 5 + 3 = 12$

Final Answer

The perimeter of the triangle is $12$ units.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would you find the area of this triangle?
2. What is the formula to calculate the distance between two points?
3. Can you describe the types of triangles based on their side lengths?
4. How would you determine if this triangle is a right triangle?
5. What happens to the perimeter if the triangle's vertices are scaled by a factor of 2?

Tip: Always double-check your calculations when dealing with square roots and distances to avoid errors.