To find the coordinates of point [tex]\( R \)[/tex], given that point [tex]\( M(5, 7) \)[/tex] is the midpoint of segment [tex]\( \overline{RS} \)[/tex] and point [tex]\( S \)[/tex] has coordinates [tex]\( (6, 9) \)[/tex], we can use the properties of midpoints.
The midpoint [tex]\( M \)[/tex] of a segment with endpoints [tex]\( R(x_1, y_1) \)[/tex] and [tex]\( S(x_2, y_2) \)[/tex] is calculated as follows:
[tex]\[
M_x = \frac{R_x + S_x}{2}
\][/tex]
[tex]\[
M_y = \frac{R_y + S_y}{2}
\][/tex]
Given:
[tex]\[
M_x = 5, \quad M_y = 7
\][/tex]
[tex]\[
S_x = 6, \quad S_y = 9
\][/tex]
We need to find the coordinates [tex]\( (R_x, R_y) \)[/tex] of point [tex]\( R \)[/tex].
1. For the x-coordinate:
[tex]\[
5 = \frac{R_x + 6}{2}
\][/tex]
To isolate [tex]\( R_x \)[/tex], multiply both sides by 2:
[tex]\[
10 = R_x + 6
\][/tex]
Then, subtract 6 from both sides:
[tex]\[
R_x = 4
\][/tex]
2. For the y-coordinate:
[tex]\[
7 = \frac{R_y + 9}{2}
\][/tex]
To isolate [tex]\( R_y \)[/tex], multiply both sides by 2:
[tex]\[
14 = R_y + 9
\][/tex]
Then, subtract 9 from both sides:
[tex]\[
R_y = 5
\][/tex]
Thus, the coordinates of [tex]\( R \)[/tex] are [tex]\( (4, 5) \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{(4, 5)}
\][/tex]
