To solve for the coordinates of point [tex]\( F \)[/tex] given the midpoint [tex]\( P \)[/tex] of segment [tex]\( \bar{EF} \)[/tex] and the coordinates of point [tex]\( E \)[/tex], follow these steps:
1. Identify the given coordinates:
- Point [tex]\( P \)[/tex] is [tex]\((-6, -2)\)[/tex]
- Point [tex]\( E \)[/tex] is [tex]\((2, -4)\)[/tex]
2. Recall the midpoint formula:
[tex]\[
P = \left(\frac{x_E + x_F}{2}, \frac{y_E + y_F}{2}\right)
\][/tex]
where [tex]\( (x_E, y_E) \)[/tex] are the coordinates of point [tex]\( E \)[/tex] and [tex]\( (x_F, y_F) \)[/tex] are the coordinates of point [tex]\( F \)[/tex].
3. Set up the equations for the midpoint coordinates:
- For the x-coordinate:
[tex]\[
-6 = \frac{2 + x_F}{2}
\][/tex]
- For the y-coordinate:
[tex]\[
-2 = \frac{-4 + y_F}{2}
\][/tex]
4. Solve for [tex]\( x_F \)[/tex] and [tex]\( y_F \)[/tex]:
- First, solve the x-coordinate equation:
[tex]\[
-6 = \frac{2 + x_F}{2}
\][/tex]
Multiply both sides by 2 to eliminate the denominator:
[tex]\[
-12 = 2 + x_F
\][/tex]
Subtract 2 from both sides:
[tex]\[
x_F = -14
\][/tex]
- Next, solve the y-coordinate equation:
[tex]\[
-2 = \frac{-4 + y_F}{2}
\][/tex]
Multiply both sides by 2 to eliminate the denominator:
[tex]\[
-4 = -4 + y_F
\][/tex]
Add 4 to both sides:
[tex]\[
y_F = 0
\][/tex]
5. Combine the results:
- The coordinates of point [tex]\( F \)[/tex] are [tex]\( (-14, 0) \)[/tex].
Thus, the coordinates of point [tex]\( F \)[/tex] are [tex]\( \boxed{(-14, 0)} \)[/tex].
