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The problem appears to involve determining which of the given points (A, B, or C) lies within a shaded half-plane that is defined by a line passing through points \( P(0, 24) \) and \( Q(18, 0) \).

### Steps to solve:

1. **Equation of the line through points P and Q**:  
   Using the formula for the slope of a line:
   \[
   \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 24}{18 - 0} = -\frac{4}{3}
   \]
   The equation of the line in slope-intercept form \( y = mx + b \) can be found. Using the point \( P(0, 24) \) for the y-intercept \( b = 24 \):
   \[
   y = -\frac{4}{3}x + 24
   \]
   
2. **Determine the half-plane**:  
   The line divides the plane into two regions. To determine which side corresponds to the shaded region, substitute a point from the shaded region (e.g., \( (0,0) \)) into the equation:
   \[
   0 \leq -\frac{4}{3}(0) + 24 \quad \text{(false)}
   \]
   So the half-plane where \( y \leq -\frac{4}{3}x + 24 \) is the region we are interested in.

3. **Test the points (A, B, C)**:  
   Plug each point into the inequality \( y \leq -\frac{4}{3}x + 24 \).

   - For \( A(777, -1020) \):
     \[
     -1020 \leq -\frac{4}{3}(777) + 24 \quad \text{(Check result)}
     \]

   - For \( B(888, -1160) \):
     \[
     -1160 \leq -\frac{4}{3}(888) + 24 \quad \text{(Check result)}
     \]

   - For \( C(999, -1300) \):
     \[
     -1300 \leq -\frac{4}{3}(999) + 24 \quad \text{(Check result)}
     \]

I will now calculate these expressions to determine which point satisfies the inequality and is part of the half-plane.Here are the results for each point when tested against the inequality \( y \leq -\frac{4}{3}x + 24 \):

- **Point A (777, -1020)**:  
  \( -1020 \leq -1012 \) — This is **true**; Point A is within the half-plane.
  
- **Point B (888, -1160)**:  
  \( -1160 \leq -1160 \) — This is **true**; Point B is also within the half-plane.
  
- **Point C (999, -1300)**:  
  \( -1300 \leq -1308 \) — This is **false**; Point C is outside the half-plane.

Both points A and B satisfy the inequality and are within the half-plane. However, since the problem asks for "which of the following" and lists only one correct answer, point **A (777, -1020)** is the best choice.

Would you like further details on how the calculations were made or have any other questions?

Here are 5 related questions to explore:
1. How do we find the equation of a line passing through two points?
2. What is the role of slope in defining a half-plane?
3. How do you test if a point lies on one side of a line in a Cartesian plane?
4. Can you generalize the method for any half-plane problem?
5. How does changing the y-intercept affect the half-plane?

**Tip:** Always verify the inequality by choosing a test point (like the origin) to ensure you're working with the correct side of the line.

Which of the points (A, B, or C) lies in the shaded half-plane defined by the line passing through P(0, 24) and Q(18, 0)? Points to test: A(777, -1020), B(888, -1160), C(999, -1300).

Solve the problem of determining which points lie in the half-plane defined by the line passing through P(0, 24) and Q(18, 0). This exercise involves coordinate geometry concepts, such as the slope formula and inequalities, and is suitable for high school students.

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