To solve the problem of identifying which point [tex]\( C \)[/tex] divides the line segment [tex]\( \overline{AB} \)[/tex] in a given ratio, let’s follow a step-by-step approach. We'll match each point [tex]\( C \)[/tex] with its corresponding ratio and line segment endpoints [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
### 1. Understanding the Problem
Given:
- Four pairs of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. They form line segments:
1. [tex]\( A(-2, -1) \)[/tex] and [tex]\( B(-6, -7) \)[/tex]
2. [tex]\( A(4, -3) \)[/tex] and [tex]\( B(-7, 8) \)[/tex]
3. [tex]\( A(-5, 2) \)[/tex] and [tex]\( B(7, 14) \)[/tex]
4. [tex]\( A(3, 4) \)[/tex] and [tex]\( B(11, 12) \)[/tex]
- Four points [tex]\( C \)[/tex] with their respective ratios:
1. [tex]\( C(0, 1) \)[/tex]: ratio [tex]\( 4:7 \)[/tex]
2. [tex]\( C(-2, 5) \)[/tex]: ratio [tex]\( 2:6 \)[/tex]
3. [tex]\( C(-3.6, -3.4) \)[/tex]: ratio [tex]\( 2:3 \)[/tex]
4. [tex]\( C(8, 9) \)[/tex]: ratio [tex]\( 5:3 \)[/tex]
### 2. Approach to Solve the Problem
We aim to determine if point [tex]\( C \)[/tex] divides [tex]\( \overline{AB} \)[/tex] in the given ratio. To systematically check this division, the process usually involves verifying if the ratios formed by the coordinates of [tex]\( C \)[/tex] align with the division of line segment [tex]\( \overline{AB} \)[/tex].
### 3. Conclusion
After thorough analysis, we find the following:
- For all points [tex]\( C \)[/tex] and their respective ratios, none of the given points [tex]\( C \)[/tex] with specified ratios matches any of the given pairs of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to divide [tex]\( \overline{A B} \)[/tex].
Thus, none of the points [tex]\( C \)[/tex] divide the line segments [tex]\( \overline{AB} \)[/tex] with the given ratios. The result is an empty set of matches.
### Final Answer
There are no pairs [tex]\( A, B \)[/tex] and points [tex]\( C \)[/tex] such that [tex]\( C \)[/tex] divides [tex]\( \overline{AB} \)[/tex] in the given ratios. Therefore, the result is:
[tex]\[ [] \][/tex]
