To determine the [tex]\( y \)[/tex]-coordinate of the midpoint of a vertical line segment with endpoints at [tex]\( (0,0) \)[/tex] and [tex]\( (0,-12) \)[/tex], we can analyze each method:
### A. Divide -12 by 2
This method is correct. The formula to find the [tex]\( y \)[/tex]-coordinate of the midpoint of a line segment with endpoints [tex]\((y_1)\)[/tex] and [tex]\((y_2)\)[/tex] is:
[tex]\[
y_{\text{midpoint}} = \frac{y_1 + y_2}{2}
\][/tex]
Plugging in [tex]\( y_1 = 0 \)[/tex] and [tex]\( y_2 = -12 \)[/tex]:
[tex]\[
y_{\text{midpoint}} = \frac{0 + (-12)}{2} = \frac{-12}{2} = -6
\][/tex]
Therefore, dividing -12 by 2 gives the correct [tex]\( y \)[/tex]-coordinate.
### B. Add the endpoints
This method is incorrect. Adding the endpoints would give us [tex]\( 0 + (-12) = -12 \)[/tex], but this does not account for averaging the distance between the two points, which is required to find the midpoint.
### C. Count by hand
While theoretically possible, this method is not precise, practical, or recommended in a formal mathematical setting. Midpoint calculations require precise arithmetic, not manual counting.
### D. Divide 12 by 2
This method is incorrect. The correct calculation should consider the negative distance, as the segment goes from [tex]\( 0 \)[/tex] to [tex]\( -12 \)[/tex]. Dividing 12 by 2 disregards the negative sign, resulting in:
[tex]\[
\frac{12}{2} = 6
\][/tex]
This would incorrectly suggest that the midpoint is at [tex]\( y = 6 \)[/tex], which is not in the correct location on the line segment, as the actual midpoint is at [tex]\( y = -6 \)[/tex].
### Conclusion
The correct method to calculate the [tex]\( y \)[/tex]-coordinate of the midpoint is:
- A. Divide -12 by 2
Hence, the valid method is Method A, and the [tex]\( y \)[/tex]-coordinate obtained is [tex]\(-6.0\)[/tex].
