Let's solve this step-by-step.
Given:
- Coordinates of point J: [tex]\((-7, 2)\)[/tex]
- Coordinates of the midpoint L: [tex]\((3, 5)\)[/tex]
We need to find the coordinates of point K. To do that, we use the midpoint formula. The midpoint [tex]\(L\)[/tex] of a line segment between two points [tex]\(J\)[/tex] [tex]\((x_1, y_1)\)[/tex] and [tex]\(K\)[/tex] [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
L\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\][/tex]
We know the coordinates of [tex]\(L\)[/tex], [tex]\(J\)[/tex], and need to find [tex]\(K\)[/tex].
First, let's set up the equations using the midpoint coordinates [tex]\((3, 5)\)[/tex]:
1. [tex]\(\frac{x_1 + x_2}{2} = 3\)[/tex]
2. [tex]\(\frac{y_1 + y_2}{2} = 5\)[/tex]
We can plug in the coordinates of [tex]\(J\)[/tex] [tex]\((-7, 2)\)[/tex]:
1. [tex]\(\frac{-7 + x_2}{2} = 3\)[/tex]
2. [tex]\(\frac{2 + y_2}{2} = 5\)[/tex]
Next, solve each of these equations for [tex]\(x_2\)[/tex] and [tex]\(y_2\)[/tex]:
1. Multiply both sides by 2:
[tex]\[
-7 + x_2 = 6
\][/tex]
Add 7 to both sides:
[tex]\[
x_2 = 13
\][/tex]
2. Multiply both sides by 2:
[tex]\[
2 + y_2 = 10
\][/tex]
Subtract 2 from both sides:
[tex]\[
y_2 = 8
\][/tex]
Therefore, the coordinates of point [tex]\(K\)[/tex] are [tex]\((13, 8)\)[/tex].
Hence, the correct answer is:
- D. [tex]\((13, 8)\)[/tex]
