To find the midpoint of a line segment with endpoints [tex]\( G(14,3) \)[/tex] and [tex]\( H(10,-6) \)[/tex], we use the midpoint formula. The midpoint formula states that the midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Given the points [tex]\( G(14,3) \)[/tex] and [tex]\( H(10,-6) \)[/tex], we can assign:
[tex]\[ x_1 = 14 \][/tex]
[tex]\[ y_1 = 3 \][/tex]
[tex]\[ x_2 = 10 \][/tex]
[tex]\[ y_2 = -6 \][/tex]
Substituting these values into the midpoint formula, we get:
[tex]\[ M = \left( \frac{14 + 10}{2}, \frac{3 + (-6)}{2} \right) \][/tex]
First, calculate the x-coordinate of the midpoint:
[tex]\[ \frac{14 + 10}{2} = \frac{24}{2} = 12 \][/tex]
Next, calculate the y-coordinate of the midpoint:
[tex]\[ \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -\frac{3}{2} \][/tex]
Therefore, the coordinates of the midpoint are:
[tex]\[ M = \left( 12, -\frac{3}{2} \right) \][/tex]
Comparing this with the given options, the correct option is:
C. [tex]\(\left(12,-\frac{3}{2}\right)\)[/tex]
