Let's break down the task from the image:

We are asked to graph the triangle formed by points $X(0, 2)$, $Y(3, -2)$, and $Z(-5, -2)$, and calculate its perimeter.

The figure shows that the correct graph is option $C$. To find the perimeter, we need to calculate the distances between the points $X$, $Y$, and $Z$ using the distance formula.

Distance Formula:

For two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance between them is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Distance $XY$:

Points: $X(0, 2)$, $Y(3, -2)$ $d_{XY} = \sqrt{(3 - 0)^2 + (-2 - 2)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

2. Distance $YZ$:

Points: $Y(3, -2)$, $Z(-5, -2)$ $d_{YZ} = \sqrt{(-5 - 3)^2 + (-2 - (-2))^2} = \sqrt{(-8)^2 + 0^2} = \sqrt{64} = 8$

3. Distance $XZ$:

Points: $X(0, 2)$, $Z(-5, -2)$ $d_{XZ} = \sqrt{(-5 - 0)^2 + (-2 - 2)^2} = \sqrt{(-5)^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \approx 6.4$

Perimeter:

Now, we sum the distances: $\text{Perimeter} = d_{XY} + d_{YZ} + d_{XZ} = 5 + 8 + 6.4 = 19.4 \text{ units}$

The perimeter of triangle $\Delta XYZ$ is approximately 19.4 units.

Would you like further details or explanations on this? Here are 5 related questions to consider:

1. How do you calculate the area of a triangle using its vertices?
2. What is the significance of the distance formula in coordinate geometry?
3. Can you describe the properties of isosceles and equilateral triangles based on side lengths?
4. How does the midpoint formula relate to the distance formula?
5. What role do slopes play in analyzing the shape of a triangle in the coordinate plane?

Tip: Always check if a triangle is right-angled using the Pythagorean theorem when you know the lengths of its sides.