Find the perimeter of the polygon with the vertices X(-1,3), and Y(3,0), and Z(-1,-2). Round your answer to the nearest hundredth.

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.

Learn how to calculate the perimeter of a polygon using coordinate geometry with vertices X(-1,3), Y(3,0), and Z(-1,-2). This problem involves applying the distance formula to find side lengths and computing the sum for the perimeter. Suitable for Grades 7-9 students.

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