To find the midpoint of the line segment with endpoints [tex]\( G(10, 1) \)[/tex] and [tex]\( H(3, 5) \)[/tex], we use the midpoint formula. The midpoint [tex]\( M(x, y) \)[/tex] of a 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]
Here, our endpoints are [tex]\( G(10, 1) \)[/tex] and [tex]\( H(3, 5) \)[/tex]. Therefore, [tex]\( x_1 = 10 \)[/tex], [tex]\( y_1 = 1 \)[/tex], [tex]\( x_2 = 3 \)[/tex], and [tex]\( y_2 = 5 \)[/tex].
Let's find the coordinates of the midpoint:
1. Calculate the [tex]\( x \)[/tex]-coordinate of the midpoint:
[tex]\[
\frac{10 + 3}{2} = \frac{13}{2} = 6.5
\][/tex]
2. Calculate the [tex]\( y \)[/tex]-coordinate of the midpoint:
[tex]\[
\frac{1 + 5}{2} = \frac{6}{2} = 3
\][/tex]
Thus, the coordinates of the midpoint are:
[tex]\[
\left( 6.5, 3.0 \right)
\][/tex]
Now, let's compare this result with the given answer choices:
A. [tex]\((-4, 9)\)[/tex]
B. [tex]\(\left( \frac{7}{2}, 2 \right)\)[/tex]
C. [tex]\(\left( \frac{13}{2}, 3 \right)\)[/tex]
D. [tex]\((13, 6)\)[/tex]
The correct answer is:
C. [tex]\(\left( \frac{13}{2}, 3 \right)\)[/tex]
