To solve the problem, we need to find the coordinates of point $E$ such that the length of segment $EC$ is one-third of the length of segment $BC$.

Step-by-step approach:

1. Identify the coordinates of points $B$ and $C$:

   • Point $B = (1, 4)$
   • Point $C = (6, -2)$

2. Calculate the distance $BC$: The formula to calculate the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Applying this to points $B(1, 4)$ and $C(6, -2)$: $BC = \sqrt{(6 - 1)^2 + (-2 - 4)^2} = \sqrt{5^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61}$ So, $BC = \sqrt{61} \approx 7.81$.

3. Find the required length of $EC$: Since $EC$ should be one-third of $BC$, we calculate: $EC = \frac{1}{3} \times BC = \frac{1}{3} \times \sqrt{61} \approx \frac{7.81}{3} \approx 2.60$

4. Determine the coordinates of $E$: To locate $E$, we find a point along the line $BC$ that divides the distance in the ratio $1:2$ (because $EC$ is one-third of $BC$). The section formula for a point dividing a line segment in the ratio $m:n$ is: $E(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right)$ where $B = (x_1, y_1)$ and $C = (x_2, y_2)$.

   In this case, $m = 1$, $n = 2$, $B(1, 4)$, and $C(6, -2)$. Substituting into the formula: $E_x = \frac{1 \cdot 6 + 2 \cdot 1}{1 + 2} = \frac{6 + 2}{3} = \frac{8}{3} \approx 2.67$ $E_y = \frac{1 \cdot (-2) + 2 \cdot 4}{1 + 2} = \frac{-2 + 8}{3} = \frac{6}{3} = 2$

   So, the coordinates of point $E$ are approximately $(2.67, 2)$.

Thus, the coordinates of point $E$ are $\left( \frac{8}{3}, 2 \right)$.

Would you like more details or have any questions about this solution? Here are some related questions:

1. How can we verify that point $E$ is correctly placed on the line $BC$?
2. How would the formula change if we wanted $EC$ to be a different fraction of $BC$?
3. What is the significance of the section formula in geometry?
4. How do we handle points on a coordinate plane that are divided in irrational ratios?
5. Can you calculate the slope of line $BC$, and how does it relate to this problem?

Tip: Always double-check your calculations by verifying distances after using the section formula!