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To solve this problem, we need to calculate the perimeter of the triangle formed by the points \((1.5, 4)\), \((4, 4)\), and \((-4, -5)\). The perimeter is the sum of the lengths of the sides of the triangle. We can find the lengths of the sides using the distance formula:

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

Let's calculate the distances for each pair of points:

1. **Distance between \((1.5, 4)\) and \((4, 4)\):**

\[
\sqrt{(4 - 1.5)^2 + (4 - 4)^2} = \sqrt{2.5^2 + 0^2} = \sqrt{6.25} = 2.5 \text{ units}
\]

2. **Distance between \((4, 4)\) and \((-4, -5)\):**

\[
\sqrt{(-4 - 4)^2 + (-5 - 4)^2} = \sqrt{(-8)^2 + (-9)^2} = \sqrt{64 + 81} = \sqrt{145} \approx 12.0 \text{ units}
\]

3. **Distance between \((1.5, 4)\) and \((-4, -5)\):**

\[
\sqrt{(-4 - 1.5)^2 + (-5 - 4)^2} = \sqrt{(-5.5)^2 + (-9)^2} = \sqrt{30.25 + 81} = \sqrt{111.25} \approx 10.5 \text{ units}
\]

Now, let's sum these distances to find the perimeter:

\[
\text{Perimeter} = 2.5 + 12.0 + 10.5 = 25.0 \text{ units}
\]

Thus, the correct answer is **25.0 units**, rounded to the nearest tenth.

Do you want more details or have any questions? 

Here are 5 related questions you might find interesting:

1. How would the perimeter change if the points were \((1.5, 4)\), \((4, 5)\), and \((-4, -5)\)?
2. Can you calculate the area of the triangle formed by these points?
3. What is the significance of using the distance formula in coordinate geometry?
4. How would the calculation change if the coordinates were in three-dimensional space?
5. Can you verify the triangle inequality theorem with these side lengths?

**Tip:** When solving geometric problems on a coordinate plane, always double-check the signs of your coordinates to avoid errors in distance calculations.

Nathan draws a triangle on the coordinate grid. He marks his points at (1.5, 4), (4, 4), and (-4, -5). What is the perimeter of the triangle, rounded to the nearest tenth of a unit?

Learn how to calculate the perimeter of a triangle on a coordinate grid with points (1.5, 4), (4, 4), and (-4, -5). This problem involves using the distance formula and is suitable for students in Grades 8-10.

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