To solve this problem, we need to reflect the point [tex]\( M (-1, -3) \)[/tex] across the line [tex]\( y = -5 \)[/tex].
We start by analyzing the distance between the point [tex]\( M \)[/tex] and the reflection line [tex]\( y = -5 \)[/tex].
1. The y-coordinate of point [tex]\( M \)[/tex] is [tex]\( -3 \)[/tex], and the y-coordinate of the reflection line is [tex]\( -5 \)[/tex].
2. The distance between [tex]\( M \)[/tex] and the reflection line [tex]\( y = -5 \)[/tex] is calculated as follows:
[tex]\[
\text{Distance} = \text{y-coordinate of } M - \text{y-coordinate of the line}
\][/tex]
[tex]\[
\text{Distance} = -3 - (-5)
\][/tex]
[tex]\[
\text{Distance} = -3 + 5
\][/tex]
[tex]\[
\text{Distance} = 2
\][/tex]
3. To find the y-coordinate of the reflected point [tex]\( M^{\prime} \)[/tex], we need to place it the same distance away from the line [tex]\( y = -5 \)[/tex], but on the opposite side.
4. Subtract this distance from the line to find the y-coordinate of [tex]\( M^{\prime} \)[/tex]:
[tex]\[
\text{y-coordinate of } M^{\prime} = \text{y-coordinate of the line} - \text{Distance}
\][/tex]
[tex]\[
\text{y-coordinate of } M^{\prime} = -5 - 2
\][/tex]
[tex]\[
\text{y-coordinate of } M^{\prime} = -7
\][/tex]
5. The x-coordinate remains unchanged during the reflection. Hence, the x-coordinate of [tex]\( M^{\prime} \)[/tex] is the same as that of [tex]\( M \)[/tex], which is [tex]\( -1 \)[/tex].
So, the coordinates of the reflected point [tex]\( M^{\prime} \)[/tex] are:
[tex]\[
M^{\prime} = (-1, -7)
\][/tex]
From the given options, the correct one is:
- [tex]\( M^{\prime}(-1, -7) \)[/tex]
Thus, the image coordinates of [tex]\( M^{\prime} \)[/tex] are indeed [tex]\( (-1, -7) \)[/tex].
