To determine the equation of the line containing the points [tex]\((5, 2)\)[/tex], [tex]\((10, 4)\)[/tex], and [tex]\((5, 6)\)[/tex], we will carry out the following steps:
1. Calculate the slopes between the points:
- Slope between [tex]\((5, 2)\)[/tex] and [tex]\((10, 4)\)[/tex]:
[tex]\[
m_1 = \frac{4 - 2}{10 - 5} = \frac{2}{5}
\][/tex]
- Slope between [tex]\((5, 2)\)[/tex] and [tex]\((5, 6)\)[/tex]:
This calculation results in a division by zero, as the x-coordinates are the same. Therefore, this line segment is vertical and the slope is undefined (or [tex]\(\infty\)[/tex]).
2. Compare the slopes:
Since one slope is [tex]\(\frac{2}{5}\)[/tex] and the other is [tex]\(\infty\)[/tex], it is clear that the points are not collinear. Thus, no single line passes through all three points.
3. Conclusion:
If the points are not collinear, they do not form a single straight line. Therefore, none of the given lines can contain all three points.
Given the analysis above, the correct conclusion is that none of the given options (A, B, or C) correctly represent a line containing all three points [tex]\((5, 2)\)[/tex], [tex]\((10, 4)\)[/tex], and [tex]\((5, 6)\)[/tex]. Hence, the result is that no valid option is available, represented numerically as [tex]\(-1\)[/tex].
