To find the coordinates of point [tex]\( A \)[/tex] given that [tex]\( M(-1, 2) \)[/tex] is the midpoint of the line segment [tex]\(\overline{AB}\)[/tex] and [tex]\( B \)[/tex] has coordinates [tex]\( (3, -5) \)[/tex], we can use the midpoint formula. The midpoint formula states that for two points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex], the coordinates of the midpoint [tex]\( M \)[/tex] are given by:
[tex]$
M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2}
$[/tex]
Given that the coordinates of [tex]\( M \)[/tex] are [tex]\( (-1, 2) \)[/tex] and the coordinates of [tex]\( B \)[/tex] are [tex]\( (3, -5) \)[/tex], we can set up two equations based on the midpoint formula.
For the [tex]\( x \)[/tex]-coordinates:
[tex]$
\frac{x_1 + 3}{2} = -1
$[/tex]
To solve for [tex]\( x_1 \)[/tex], we multiply both sides by 2:
[tex]$
x_1 + 3 = -2
$[/tex]
Then, subtract 3 from both sides:
[tex]$
x_1 = -2 - 3
$[/tex]
[tex]$
x_1 = -5
$[/tex]
For the [tex]\( y \)[/tex]-coordinates:
[tex]$
\frac{y_1 + (-5)}{2} = 2
$[/tex]
To solve for [tex]\( y_1 \)[/tex], we multiply both sides by 2:
[tex]$
y_1 - 5 = 4
$[/tex]
Then, add 5 to both sides:
[tex]$
y_1 = 4 + 5
$[/tex]
[tex]$
y_1 = 9
$[/tex]
Therefore, the coordinates of point [tex]\( A \)[/tex] are [tex]\( (-5, 9) \)[/tex].
