The problem in the image states that a line segment $VW$ is divided by point $U$ such that the ratio $VU:UW$ is $3:1$. Your task is to graph point $U$ on the given line. Let's go through the solution:

1. Identifying Coordinates of Points V and W:

   • Point $V$ is at approximately $(2, 4)$.
   • Point $W$ is at approximately $(14, 18)$.

2. Using the Section Formula: Since $U$ divides the line in the ratio $3:1$, we apply the section formula for internal division: $U(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)$ where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of points $V$ and $W$, and $m:n$ is the given ratio.

   Here, $m = 3$ and $n = 1$, and the coordinates of $V$ are $(2, 4)$ and $W$ are $(14, 18)$.

3. Calculating Coordinates of U: $U(x, y) = \left( \frac{3 \cdot 14 + 1 \cdot 2}{3+1}, \frac{3 \cdot 18 + 1 \cdot 4}{3+1} \right)$ Simplifying the calculations: $U(x, y) = \left( \frac{42 + 2}{4}, \frac{54 + 4}{4} \right) = \left( \frac{44}{4}, \frac{58}{4} \right) = (11, 14.5)$

Thus, point $U$ has coordinates $(11, 14.5)$. You can graph it at that location on the given grid.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you calculate the midpoint of a line segment?
2. Can the section formula be used if $U$ divides the line externally?
3. What is the distance formula between two points on a plane?
4. How would the coordinates change if the ratio $VU:UW$ were $2:1$?
5. How do you find the slope of a line given two points?

Tip: When dealing with line division problems, always ensure to correctly identify the ratio and whether the division is internal or external to avoid calculation errors.