To find the coordinates of point [tex]\( K \)[/tex] given that the midpoint of segment [tex]\( \overline{J K} \)[/tex] is [tex]\( L(3,5) \)[/tex] and the coordinates of point [tex]\( J \)[/tex] are [tex]\((-7,2)\)[/tex], we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint [tex]\( (x_{\text{mid}}, y_{\text{mid}}) \)[/tex] can be found using the coordinates of the endpoints [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] as follows:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
We know point [tex]\( J(-7,2) \)[/tex] and the midpoint [tex]\( L(3,5) \)[/tex]. Plugging these into the midpoint formula gives us:
[tex]\[
\left( \frac{-7 + x_2}{2}, \frac{2 + y_2}{2} \right) = (3, 5)
\][/tex]
To solve for [tex]\( x_2 \)[/tex] and [tex]\( y_2 \)[/tex], we set up the following equations from the equality of the coordinates:
[tex]\[
\frac{-7 + x_2}{2} = 3
\][/tex]
[tex]\[
\frac{2 + y_2}{2} = 5
\][/tex]
Now, solve these equations step-by-step. For the [tex]\( x \)[/tex]-coordinate:
[tex]\[
\frac{-7 + x_2}{2} = 3
\][/tex]
Multiply both sides by 2 to eliminate the denominator:
[tex]\[
-7 + x_2 = 6
\][/tex]
Add 7 to both sides to solve for [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = 13
\][/tex]
Next, for the [tex]\( y \)[/tex]-coordinate:
[tex]\[
\frac{2 + y_2}{2} = 5
\][/tex]
Multiply both sides by 2 to eliminate the denominator:
[tex]\[
2 + y_2 = 10
\][/tex]
Subtract 2 from both sides to solve for [tex]\( y_2 \)[/tex]:
[tex]\[
y_2 = 8
\][/tex]
Thus, the coordinates of point [tex]\( K \)[/tex] are [tex]\( (13, 8) \)[/tex].
The correct answer is:
A. [tex]\( (13, 8) \)[/tex]
