Let's find the coordinates of point \( K \) given that the coordinates of point \( J \) are \( (-7,2) \) and the midpoint \( L \) of \( \overline{JK} \) is at \( (3,5) \).
To solve this, we use the midpoint formula. The midpoint formula states that the coordinates of the midpoint \( L \) of a segment with endpoints \( J(x_1, y_1) \) and \( K(x_2, y_2) \) are given by:
[tex]\[ L_x = \frac{x_1 + x_2}{2} \][/tex]
[tex]\[ L_y = \frac{y_1 + y_2}{2} \][/tex]
Given:
- The coordinates of \( J(x_1, y_1) \) = \((-7,2)\)
- The coordinates of midpoint \( L(L_x, L_y) \) = \((3,5)\)
We need to find the coordinates of \( K(x_2, y_2) \).
Since we know \( L_x \) is the average of \( x_1 \) and \( x_2 \):
[tex]\[ 3 = \frac{-7 + x_2}{2} \][/tex]
Multiplying both sides by 2 to solve for \( x_2 \):
[tex]\[ 6 = -7 + x_2 \][/tex]
[tex]\[ x_2 = 6 + 7 \][/tex]
[tex]\[ x_2 = 13 \][/tex]
Similarly, for \( L_y \):
[tex]\[ 5 = \frac{2 + y_2}{2} \][/tex]
Multiplying both sides by 2 to solve for \( y_2 \):
[tex]\[ 10 = 2 + y_2 \][/tex]
[tex]\[ y_2 = 10 - 2 \][/tex]
[tex]\[ y_2 = 8 \][/tex]
Therefore, the coordinates of point \( K \) are \((13, 8)\).
So, the correct answer is:
A. [tex]\((13,8)\)[/tex]
