https://file.aiquickdraw.com/mathbot/file/1726615850598gs203j7s.jpg

To solve this problem, we need to find the coordinates of point \( Z \) that divides the line segment \( ST \) in the ratio \( SZ : ZT = 3 : 1 \).

### Step 1: Coordinates of \( S \) and \( T \)
From the graph, we observe that the coordinates of the points are:
- \( S(4, 1) \)
- \( T(0, 3) \)

### Step 2: Formula for the Section (Partition) of a Line
The formula for finding a point \( Z(x, y) \) that divides a line segment in a ratio \( m : n \) between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:

\[
Z(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)
\]

Here, \( m = 3 \), \( n = 1 \), \( S(4, 1) \), and \( T(0, 3) \).

### Step 3: Applying the Formula
Using the formula, we can calculate the coordinates of \( Z \):

\[
x_Z = \frac{3 \cdot 0 + 1 \cdot 4}{3+1} = \frac{4}{4} = 1
\]
\[
y_Z = \frac{3 \cdot 3 + 1 \cdot 1}{3+1} = \frac{9 + 1}{4} = \frac{10}{4} = 2.5
\]

So, the coordinates of \( Z \) are \( Z(1, 2.5) \).

### Step 4: Identifying Closest Option
The closest answer to this result is not listed exactly, but based on the graph, we can deduce the point closest to this might be:

- \( Z(1, 2) \).

Would you like more details or have any questions?

### Here are 5 related questions:
1. What happens if the ratio changes to \( SZ : ZT = 2 : 1 \)?
2. How do we find the midpoint of a line segment?
3. Can this method be applied in three dimensions? If yes, how?
4. What would the coordinates of \( Z \) be if the ratio was reversed to \( 1 : 3 \)?
5. How can we use vector methods to solve this problem?

### Tip:
When dividing a line segment in a given ratio, always double-check your points and the correct order of the ratio.

Analyze the graph of the line segment ST below and identify the point Z that would partition ST into the ratio SZ : ZT = 3 : 1.

Learn how to find the coordinates of point Z that partitions a line segment ST in the ratio SZ:ZT = 3:1. Apply the section formula and solve geometry problems easily. Suitable for students in Grades 9-10.

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