Find the perimeter of the triangle whose vertices are (−5,−4)(−5,−4), (7,−4)(7,−4), and (7,1)(7,1). Write the exact answer. Do not round.

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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Do you want further details or have any other questions?

### 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.

Learn how to calculate the perimeter of a triangle using the distance formula with vertices at (-5, -4), (7, -4), and (7, 1). Detailed step-by-step solution included. Suitable for students in Grades 9-12.

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