To find the midpoints of each of the sides [tex]\(\overline{JM}\)[/tex] and [tex]\(\overline{KL}\)[/tex] in quadrilateral [tex]\(J K L M\)[/tex] with vertices [tex]\(J(-7, -2)\)[/tex], [tex]\(K(-2, -2)\)[/tex], [tex]\(L(-3, -4)\)[/tex], and [tex]\(M(-6, -4)\)[/tex], we need to follow these steps:
1. Identify the coordinates of the vertices:
- [tex]\(J(-7, -2)\)[/tex]
- [tex]\(M(-6, -4)\)[/tex]
- [tex]\(K(-2, -2)\)[/tex]
- [tex]\(L(-3, -4)\)[/tex]
2. Calculate the midpoint of [tex]\(\overline{JM}\)[/tex]:
- To find the midpoint [tex]\((x_m, y_m)\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], use the midpoint formula:
[tex]\[
x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}
\][/tex]
- Applying this to [tex]\(\overline{JM}\)[/tex] where [tex]\(J(-7, -2)\)[/tex] and [tex]\(M(-6, -4)\)[/tex]:
[tex]\[
x_{JM} = \frac{-7 + (-6)}{2} = \frac{-13}{2} = -6.5
\][/tex]
[tex]\[
y_{JM} = \frac{-2 + (-4)}{2} = \frac{-6}{2} = -3
\][/tex]
- The midpoint of [tex]\(\overline{JM}\)[/tex] is [tex]\((-6.5, -3)\)[/tex].
3. Calculate the midpoint of [tex]\(\overline{KL}\)[/tex]:
- Using the same midpoint formula for [tex]\(\overline{KL}\)[/tex] where [tex]\(K(-2, -2)\)[/tex] and [tex]\(L(-3, -4)\)[/tex]:
[tex]\[
x_{KL} = \frac{-2 + (-3)}{2} = \frac{-5}{2} = -2.5
\][/tex]
[tex]\[
y_{KL} = \frac{-2 + (-4)}{2} = \frac{-6}{2} = -3
\][/tex]
- The midpoint of [tex]\(\overline{KL}\)[/tex] is [tex]\((-2.5, -3)\)[/tex].
So, the midpoints of the sides [tex]\(\overline{JM}\)[/tex] and [tex]\(\overline{KL}\)[/tex] are [tex]\((-6.5, -3)\)[/tex] and [tex]\((-2.5, -3)\)[/tex], respectively.
Thus, the correct answer is:
D. [tex]\( M_{\frac{JM}{JM}}(-6.5, -3); M_{\frac{KL}{KL}}(-2.5, -3) \)[/tex].
