To solve the problem of finding the perimeter of the polygon formed by the points $U(-2, 4)$, $V(3, 4)$, and $W(3, -4)$, we need to calculate the distances between each pair of points and then sum them to find the perimeter.

Steps:

1. Calculate the distance between points $U$ and $V$: Using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ For points $U(-2, 4)$ and $V(3, 4)$: $d(U, V) = \sqrt{(3 - (-2))^2 + (4 - 4)^2} = \sqrt{5^2 + 0^2} = \sqrt{25} = 5$

2. Calculate the distance between points $V$ and $W$: For points $V(3, 4)$ and $W(3, -4)$: $d(V, W) = \sqrt{(3 - 3)^2 + (-4 - 4)^2} = \sqrt{0^2 + (-8)^2} = \sqrt{64} = 8$

3. Calculate the distance between points $W$ and $U$: For points $W(3, -4)$ and $U(-2, 4)$: $d(W, U) = \sqrt{(-2 - 3)^2 + (4 - (-4))^2} = \sqrt{(-5)^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.43$

4. Find the perimeter: The perimeter is the sum of these distances: $\text{Perimeter} = 5 + 8 + 9.43 = 22.43$

So, the perimeter of the polygon is approximately 22.43 units.

Do you want more details or have any questions?

Here are some related questions:

1. How is the distance formula derived from the Pythagorean theorem?
2. What are the general steps to calculate the perimeter of any polygon?
3. How can the coordinates of points help in determining the shape of a polygon?
4. What are some real-life applications of finding the perimeter?
5. How would the perimeter change if one of the coordinates was different?

Tip: Always double-check the coordinates of points when calculating distances in geometry problems to avoid errors in distance calculations.