To find the midpoint of a line segment [tex]\(\overline{GH}\)[/tex] with endpoints [tex]\(G(10, 1)\)[/tex] and [tex]\(H(3, 5)\)[/tex], follow these steps:
1. Identify the given coordinates:
- [tex]\(G(10, 1)\)[/tex]: The coordinates of point G.
- [tex]\(H(3, 5)\)[/tex]: The coordinates of point H.
2. Use the midpoint formula:
The formula to find the midpoint of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
3. Substitute the coordinates of points G and H into the formula:
- [tex]\(x_1 = 10\)[/tex], [tex]\(y_1 = 1\)[/tex]
- [tex]\(x_2 = 3\)[/tex], [tex]\(y_2 = 5\)[/tex]
4. Calculate the x-coordinate of the midpoint:
[tex]\[
\frac{10 + 3}{2} = \frac{13}{2} = 6.5
\][/tex]
5. Calculate the y-coordinate of the midpoint:
[tex]\[
\frac{1 + 5}{2} = \frac{6}{2} = 3
\][/tex]
6. Combine the results to find the midpoint:
The midpoint of [tex]\(\overline{GH}\)[/tex] is [tex]\((6.5, 3)\)[/tex].
7. Check the provided 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]
Our computation yields [tex]\((6.5, 3)\)[/tex], which matches choice C if rewritten as [tex]\(\left(\frac{13}{2}, 3\right)\)[/tex] (since [tex]\(6.5\)[/tex] is equal to [tex]\(\frac{13}{2}\)[/tex]).
Hence, the correct answer is:
C. [tex]\(\left(\frac{13}{2}, 3\right)\)[/tex]
