To find the perimeter of the triangle with vertices $U(-2, 4)$, $V(3, 4)$, and $W(3, -4)$, we need to calculate the lengths of each side of the triangle and then sum them up.

Step 1: Calculate the length of each side using the distance formula

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

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

1. Length of $UV$: $U(-2, 4) \text{ and } V(3, 4)$ $UV = \sqrt{(3 - (-2))^2 + (4 - 4)^2} = \sqrt{(3 + 2)^2 + 0^2} = \sqrt{5^2} = 5$

2. Length of $VW$: $V(3, 4) \text{ and } W(3, -4)$ $VW = \sqrt{(3 - 3)^2 + (-4 - 4)^2} = \sqrt{0^2 + (-8)^2} = \sqrt{64} = 8$

3. Length of $UW$: $U(-2, 4) \text{ and } W(3, -4)$ $UW = \sqrt{(3 - (-2))^2 + (-4 - 4)^2} = \sqrt{(3 + 2)^2 + (-8)^2} = \sqrt{5^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89}$

Step 2: Calculate the perimeter

The perimeter $P$ of the triangle is the sum of the lengths of its sides:

$P = UV + VW + UW = 5 + 8 + \sqrt{89}$

$P = 13 + \sqrt{89}$

Final Answer

The perimeter of the triangle is $13 + \sqrt{89}$.

Would you like further details or have any questions?

Here are five related questions to further explore this topic:

1. How do you find the area of the triangle given its vertices?
2. Can the triangle's type (e.g., scalene, isosceles) be determined from the given vertices?
3. How can you find the centroid of the triangle with the given vertices?
4. What is the formula for finding the circumradius of a triangle using its vertices?
5. How would the perimeter change if one of the triangle's vertices were moved?

Tip: When dealing with geometry problems involving coordinates, always sketch the points and label them on the coordinate plane for a visual understanding of the problem.