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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!

Create point E such that EC is one-third the distance of BC.

Learn how to solve a geometry problem involving points on a coordinate plane. This problem involves calculating the distance between points using the distance formula and dividing a line segment using the section formula. Suitable for students in Grades 8-10.

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