To find the perimeter of the triangle with vertices $(-5, -4)$, $(7, -4)$, and $(7, 1)$, we need to calculate the lengths of all three sides using the distance formula, and then sum these lengths.

Step 1: Calculate the Lengths of the Sides

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

1. Length of side between $(-5, -4)$ and $(7, -4)$: $d_1 = \sqrt{(7 - (-5))^2 + (-4 - (-4))^2} = \sqrt{(7 + 5)^2 + 0^2} = \sqrt{12^2} = 12$

2. Length of side between $(7, -4)$ and $(7, 1)$: $d_2 = \sqrt{(7 - 7)^2 + (1 - (-4))^2} = \sqrt{0 + (1 + 4)^2} = \sqrt{5^2} = 5$

3. Length of side between $(7, 1)$ and $(-5, -4)$: $d_3 = \sqrt{((-5) - 7)^2 + (-4 - 1)^2} = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13$

Step 2: Calculate the Perimeter

The perimeter $P$ of the triangle is the sum of the lengths of all three sides:

$P = d_1 + d_2 + d_3 = 12 + 5 + 13 = 30$

Final Answer:

The perimeter of the triangle is 30 units.

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Related Questions:

1. How can you determine if a triangle is a right triangle using the coordinates of its vertices?
2. What is the formula for calculating the area of a triangle given its side lengths (Heron's formula)?
3. How do you find the midpoint of a side in a triangle?
4. What is the relationship between the perimeter and the area of a triangle?
5. How can you verify if three points are collinear using the distance formula?

Tip:

Always double-check your arithmetic when using the distance formula, especially when dealing with negative coordinates, to avoid errors.