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Select the correct answer.

The endpoints of [tex]\overline{GH}[/tex] are [tex]G(14,3)[/tex] and [tex]H(10,-6)[/tex]. What is the midpoint of [tex]\overline{GH}[/tex]?

A. [tex](6, -15)[/tex]
B. [tex]\left(-2, -\frac{9}{2}\right)[/tex]
C. [tex]\left(12, -\frac{3}{2}\right)[/tex]
D. [tex](24, -3)[/tex]
E. [tex](18, 12)[/tex]


Sagot :

To determine the midpoint of a line segment, we use the midpoint formula. The midpoint [tex]\((M_x, M_y)\)[/tex] of a line segment with endpoints [tex]\(G(x_1, y_1)\)[/tex] and [tex]\(H(x_2, y_2)\)[/tex] is given by:

[tex]\[ M_x = \frac{x_1 + x_2}{2} \][/tex]
[tex]\[ M_y = \frac{y_1 + y_2}{2} \][/tex]

Given the endpoints [tex]\(G(14, 3)\)[/tex] and [tex]\(H(10, -6)\)[/tex], we can substitute the coordinates into the midpoint formula.

First, calculate the x-coordinate of the midpoint:

[tex]\[ M_x = \frac{14 + 10}{2} = \frac{24}{2} = 12 \][/tex]

Next, calculate the y-coordinate of the midpoint:

[tex]\[ M_y = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -\frac{3}{2} \][/tex]

So, the midpoint of [tex]\(\overline{GH}\)[/tex] is:

[tex]\[ \left(12, -\frac{3}{2}\right) \][/tex]

Therefore, the correct answer is:

[tex]\[ \boxed{\left(12, -\frac{3}{2}\right)} \][/tex]