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To solve this problem, let’s go through each part step-by-step.

1. **Identifying Points and Ratios**: 
   - Given points: \( A(6, 0) \), \( B(0, 3) \), \( C(-6, 0) \), and \( D(0, -3) \).
   - We need to find the coordinates of a point \( I \) that divides line segment \( AB \) in the ratio \( 2:1 \).

   The coordinates of a point that divides a segment in a given ratio are calculated by:
   \[
   x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \quad \text{and} \quad y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
   \]
   For point \( I \), substituting \( A(6,0) \) and \( B(0,3) \), we have \( m = 2 \) and \( n = 1 \):
   \[
   x = \frac{2 \cdot 0 + 1 \cdot 6}{2 + 1} = \frac{6}{3} = 2, \quad y = \frac{2 \cdot 3 + 1 \cdot 0}{2 + 1} = \frac{6}{3} = 2
   \]
   So, \( I \) has coordinates \( (2, 2) \).

2. **Equation of Line \( BC \)**:
   - The slope of \( BC \) is:
     \[
     \text{slope} = \frac{0 - 3}{-6 - 0} = \frac{-3}{-6} = \frac{1}{2}
     \]
   - Using the point-slope form \( y = mx + c \) with point \( B(0,3) \):
     \[
     y = \frac{1}{2}x + 3
     \]
   So, the equation of line \( BC \) is \( y = \frac{1}{2}x + 3 \).

3. **Equation of Line \( HI \)**:
   - Line \( HI \) is perpendicular to \( BC \), so its slope is the negative reciprocal of \( \frac{1}{2} \), which is \( -2 \).
   - Using point \( I(2, 2) \), the equation for \( HI \) becomes:
     \[
     y - 2 = -2(x - 2)
     \]
     Expanding:
     \[
     y = -2x + 4 + 2 = -2x + 6
     \]
   So, the equation of \( HI \) is \( y = -2x + 6 \).

4. **Checking Point H and Circle \( K \)**:
   - Next steps would involve finding point \( H \), dividing lines, and confirming point locations relative to circle \( K \), but let’s confirm if you want further details on these calculations.

Would you like further details on each step or help with the remaining parts?

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Here are some related questions for deeper understanding:

1. How do you calculate the midpoint of a line segment?
2. What is the process to determine if a point lies on a given line?
3. How can we verify if two lines are perpendicular?
4. How do we find the equation of a circle given a center and a radius?
5. What is the importance of ratios in dividing a line segment?

**Tip**: Always check your slopes when working with perpendicular lines—remember, perpendicular slopes are negative reciprocals!

Given points A(6, 0), B(0, 3), C(-6, 0), and D(0, -3), find the point I that divides line segment AB in a 2:1 ratio, the equation of line BC, and details about the circle K related to the midpoint of a line segment.

Learn to solve a coordinate geometry problem involving points A(6, 0), B(0, 3), C(-6, 0), and D(0, -3). This problem covers dividing line segments in a ratio, finding line equations, and exploring circle properties. Suitable for Grades 10-12.

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