To determine the coordinates of point [tex]\( B \)[/tex] given the midpoint [tex]\( M \)[/tex] of a line segment [tex]\( AB \)[/tex] and one endpoint [tex]\( A \)[/tex], follow these steps:
1. First, recall the midpoint formula for a line segment. If [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] are the endpoints of the line segment, the coordinates of the midpoint [tex]\( M \)[/tex] are:
[tex]\[
M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
2. Here, we are given:
- The coordinates of midpoint [tex]\( M \)[/tex] are [tex]\( (3, 7) \)[/tex].
- The coordinates of point [tex]\( A \)[/tex] are [tex]\( (6, 8) \)[/tex].
3. Let the coordinates of point [tex]\( B \)[/tex] be [tex]\( (x_2, y_2) \)[/tex].
4. Using the midpoint formula, set up the equations for the [tex]\( x \)[/tex]-coordinate and [tex]\( y \)[/tex]-coordinate separately:
[tex]\[
3 = \frac{6 + x_2}{2}
\][/tex]
[tex]\[
7 = \frac{8 + y_2}{2}
\][/tex]
5. To find [tex]\( x_2 \)[/tex], solve the first equation:
[tex]\[
3 = \frac{6 + x_2}{2}
\][/tex]
Multiply both sides by 2 to eliminate the fraction:
[tex]\[
6 = 6 + x_2
\][/tex]
Subtract 6 from both sides:
[tex]\[
0 = x_2
\][/tex]
So, the [tex]\( x \)[/tex]-coordinate of [tex]\( B \)[/tex] is [tex]\( 0 \)[/tex].
6. Next, solve for [tex]\( y_2 \)[/tex] using the second equation:
[tex]\[
7 = \frac{8 + y_2}{2}
\][/tex]
Multiply both sides by 2 to eliminate the fraction:
[tex]\[
14 = 8 + y_2
\][/tex]
Subtract 8 from both sides:
[tex]\[
6 = y_2
\][/tex]
So, the [tex]\( y \)[/tex]-coordinate of [tex]\( B \)[/tex] is [tex]\( 6 \)[/tex].
7. Therefore, the coordinates of point [tex]\( B \)[/tex] are [tex]\( (0, 6) \)[/tex].
In summary, the coordinates of point [tex]\( B \)[/tex] are [tex]\((0, 6)\)[/tex].
