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The problem involves finding the coordinates of point \( B \) on line segment \( \overline{AC} \) such that \( AB \) is 3 times as long as \( BC \). Let's solve this step-by-step using the coordinates of points \( A \) and \( C \) given in the graph.

### Step 1: Identify coordinates of \( A \) and \( C \)
From the graph:
- Point \( A \) is at \( (4, 5) \)
- Point \( C \) is at \( (2, 0) \)

### Step 2: Use the section formula
Since \( AB \) is 3 times as long as \( BC \), the ratio of the distances is \( AB:BC = 3:1 \). The section formula gives the coordinates of a point that divides a line segment in a given ratio.

The section formula for dividing a line segment between points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) is:

\[
(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Here, we want to divide \( \overline{AC} \) in the ratio \( 3:1 \), with:
- \( A = (4, 5) \)
- \( C = (2, 0) \)

### Step 3: Apply the section formula
Plugging the values into the formula, with \( m = 3 \) and \( n = 1 \):

\[
x_B = \frac{3 \cdot 2 + 1 \cdot 4}{3 + 1} = \frac{6 + 4}{4} = \frac{10}{4} = 2.5
\]

\[
y_B = \frac{3 \cdot 0 + 1 \cdot 5}{3 + 1} = \frac{0 + 5}{4} = \frac{5}{4} = 1.25
\]

### Step 4: Conclusion
The coordinates of point \( B \) are \( (2.5, 1.25) \).

Would you like a deeper explanation of any steps or additional details?

Here are 5 related questions for further practice:
1. How would you calculate the midpoint of \( \overline{AC} \)?
2. If \( AB \) were twice as long as \( BC \), what would the coordinates of \( B \) be?
3. Can you find the slope of line segment \( AC \)?
4. How would you calculate the distance between points \( A \) and \( C \)?
5. What would the coordinates of \( B \) be if \( AB = BC \) (equal length)?

**Tip:** The section formula is useful for dividing line segments into specific ratios, which is especially handy in coordinate geometry.

What are the coordinates of point B on line AC such that AB is 3 times as long as BC?

Learn how to find the coordinates of point B on line segment AC such that AB is 3 times as long as BC. This problem demonstrates the use of the section formula to divide a line segment in a given ratio. Suitable for students in Grades 8-10.

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