To find the perimeter of the triangle with vertices $(-4,-7)$, $(12,5)$, and $(3,-7)$, we need to calculate the lengths of its three sides using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 1: Find the side lengths

Side 1: Between $(-4,-7)$ and $(12,5)$

$d_1 = \sqrt{(12 - (-4))^2 + (5 - (-7))^2}$

$= \sqrt{(12 + 4)^2 + (5 + 7)^2}$

$= \sqrt{16^2 + 12^2}$

$= \sqrt{256 + 144}$

$= \sqrt{400} = 20$

Side 2: Between $(12,5)$ and $(3,-7)$

$d_2 = \sqrt{(3 - 12)^2 + (-7 - 5)^2}$

$= \sqrt{(-9)^2 + (-12)^2}$

$= \sqrt{81 + 144}$

$= \sqrt{225} = 15$

Side 3: Between $(3,-7)$ and $(-4,-7)$

$d_3 = \sqrt{(-4 - 3)^2 + (-7 - (-7))^2}$

$= \sqrt{(-7)^2 + 0^2}$

$= \sqrt{49} = 7$

Step 2: Find the Perimeter

$P = d_1 + d_2 + d_3$

$= 20 + 15 + 7$

$= 42$

Final Answer:

$\boxed{42}$

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Would you like a step-by-step breakdown of a specific part?

Here are some related questions you might find useful:

1. How do you derive the distance formula?
2. How do you find the area of the same triangle using determinants?
3. How do you determine if the given triangle is right-angled?
4. How do you find the midpoint of a segment in a coordinate plane?
5. How do you determine the equation of a line passing through two given points?

Tip: Always check whether a triangle is right-angled by using the Pythagorean theorem with its side lengths!