To find the perimeter of the polygon formed by the vertices $X(-1,3)$, $Y(3,0)$, and $Z(-1,-2)$, we need to calculate the distances between each pair of points (i.e., the lengths of the sides of the triangle) and then sum these distances.

Step 1: Calculate the distances between the points

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

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

1. Distance between $X(-1,3)$ and $Y(3,0)$:

$XY = \sqrt{(3 - (-1))^2 + (0 - 3)^2} = \sqrt{(3 + 1)^2 + (-3)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

2. Distance between $Y(3,0)$ and $Z(-1,-2)$:

$YZ = \sqrt{(-1 - 3)^2 + (-2 - 0)^2} = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47$

3. Distance between $Z(-1,-2)$ and $X(-1,3)$:

$ZX = \sqrt{(-1 - (-1))^2 + (3 - (-2))^2} = \sqrt{(0)^2 + (5)^2} = \sqrt{0 + 25} = \sqrt{25} = 5$

Step 2: Calculate the perimeter

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

$P = XY + YZ + ZX = 5 + 4.47 + 5 = 14.47$

Final Answer:

The perimeter of the polygon is approximately 14.47 units.

Would you like more details or have any questions?

Here are 5 questions that expand on the information:

1. How do you calculate the area of a triangle given its vertices?
2. What is the difference between perimeter and area?
3. Can the perimeter of a polygon ever be negative? Why or why not?
4. How does rounding affect the accuracy of geometric measurements?
5. What are some real-world applications of calculating the perimeter of shapes?

Tip: When working with distances on a coordinate plane, always double-check the sign of your coordinates, as a small mistake can lead to incorrect distance calculations.