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.