To find the length of the median from vertex [tex]\( C \)[/tex] in triangle [tex]\( ABC \)[/tex], we need to follow two main steps: first, determine the midpoint of side [tex]\( AB \)[/tex], and second, calculate the distance from [tex]\( C \)[/tex] to this midpoint.
1. Finding the midpoint of [tex]\( AB \)[/tex]:
The vertices of the triangle are:
- [tex]\( A(-8, 8) \)[/tex]
- [tex]\( B(6, 2) \)[/tex]
The coordinates of the midpoint [tex]\( M \)[/tex] of [tex]\( AB \)[/tex] are given by the midpoint formula:
[tex]\[ M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \][/tex]
Substituting in the given coordinates for [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ M_x = \frac{-8 + 6}{2} = \frac{-2}{2} = -1 \][/tex]
[tex]\[ M_y = \frac{8 + 2}{2} = \frac{10}{2} = 5 \][/tex]
So the midpoint [tex]\( M \)[/tex] of [tex]\( AB \)[/tex] is [tex]\((-1, 5)\)[/tex].
2. Finding the length of the median from [tex]\( C \)[/tex] to [tex]\( M \)[/tex]:
The vertex [tex]\( C \)[/tex] has coordinates [tex]\((-2, 1)\)[/tex].
The distance [tex]\( CM \)[/tex] is calculated using the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] are the coordinates of [tex]\( C \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of the midpoint [tex]\( M \)[/tex].
Substituting in the coordinates of [tex]\( C \)[/tex] and [tex]\( M \)[/tex]:
[tex]\[ d = \sqrt{(-1 - (-2))^2 + (5 - 1)^2} \][/tex]
[tex]\[ d = \sqrt{(1)^2 + (4)^2} \][/tex]
[tex]\[ d = \sqrt{1 + 16} \][/tex]
[tex]\[ d = \sqrt{17} \][/tex]
Thus, the length of the median from [tex]\( \angle C \)[/tex] in triangle [tex]\( ABC \)[/tex] is [tex]\( \sqrt{17} \)[/tex], which is approximately equal to
Since the closest option accurately reflecting our computed length [tex]\( \sqrt{17} \)[/tex] is not among the provided options, answer choice D seems the most appropriate. If we assume 'sqrt{} 65' is a typographical error, none of the given options correspond directly to [tex]\(\sqrt{17}\)[/tex].
Considering the mistake in rephrasing options D correctly:
- Correct length of median: [tex]\( \sqrt{17} \approx 4.123 \)[/tex]
Hence, picking the most logical closest it seems, -> would be approximately 4
Upon reasonable math considering no entry extend finding an accurate balance altogether.
Thus, the closest correct approximation would be:
[tex]\[ B. \quad 4 \][/tex]
Always ensure to validate final answer keeping arithmetic intact.
