To determine the coordinates of point [tex]\( B \)[/tex] given that [tex]\( M(3, 4) \)[/tex] is the midpoint of the line segment [tex]\( AB \)[/tex] and [tex]\( A(-1, 6) \)[/tex], we can use the midpoint formula. The midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] is given by:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Given that the coordinates of point [tex]\( M \)[/tex] are [tex]\( (3, 4) \)[/tex], we can set up the following equations based on the formula:
[tex]\[
3 = \frac{-1 + x_2}{2}
\][/tex]
[tex]\[
4 = \frac{6 + y_2}{2}
\][/tex]
To solve for [tex]\( x_2 \)[/tex] and [tex]\( y_2 \)[/tex], we first clear the fractions by multiplying both sides of each equation by 2:
[tex]\[
2 \times 3 = -1 + x_2 \implies 6 = -1 + x_2 \implies x_2 = 7
\][/tex]
[tex]\[
2 \times 4 = 6 + y_2 \implies 8 = 6 + y_2 \implies y_2 = 2
\][/tex]
Thus, the coordinates of point [tex]\( B \)[/tex] are [tex]\( (7, 2) \)[/tex].
Therefore, the correct answer is:
a) [tex]\((7, 2)\)[/tex]
