Answer :
To find the midpoint of a line segment [tex]\(\overline{GH}\)[/tex] given the endpoints [tex]\(G(14, 3)\)[/tex] and [tex]\(H(10, -6)\)[/tex], we use the midpoint formula. The midpoint formula states that the coordinates of the midpoint [tex]\(M\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Let's apply this formula step by step:
1. Identify the coordinates of the endpoints:
- [tex]\(G(14, 3)\)[/tex]: Here, [tex]\(x_1 = 14\)[/tex] and [tex]\(y_1 = 3\)[/tex].
- [tex]\(H(10, -6)\)[/tex]: Here, [tex]\(x_2 = 10\)[/tex] and [tex]\(y_2 = -6\)[/tex].
2. Calculate the x-coordinate of the midpoint:
[tex]\[ \text{midpoint}_x = \frac{x_1 + x_2}{2} = \frac{14 + 10}{2} = \frac{24}{2} = 12 \][/tex]
3. Calculate the y-coordinate of the midpoint:
[tex]\[ \text{midpoint}_y = \frac{y_1 + y_2}{2} = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -1.5 \][/tex]
Therefore, the coordinates of the midpoint [tex]\(M\)[/tex] are:
[tex]\[ M = (12, -1.5) \][/tex]
Given the answer choices:
- A. [tex]\((6, -15)\)[/tex]
- B. [tex]\(\left(-2, -\frac{2}{2}\right)\)[/tex]
- C. [tex]\(\left(12, -\frac{3}{2}\right)\)[/tex]
- D. [tex]\((24, -3)\)[/tex]
- E. [tex]\((18, 12)\)[/tex]
The correct answer is:
C. [tex]\(\left(12, -\frac{3}{2}\right)\)[/tex]
This matches the coordinates [tex]\((12, -1.5)\)[/tex] obtained from our calculations.
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Let's apply this formula step by step:
1. Identify the coordinates of the endpoints:
- [tex]\(G(14, 3)\)[/tex]: Here, [tex]\(x_1 = 14\)[/tex] and [tex]\(y_1 = 3\)[/tex].
- [tex]\(H(10, -6)\)[/tex]: Here, [tex]\(x_2 = 10\)[/tex] and [tex]\(y_2 = -6\)[/tex].
2. Calculate the x-coordinate of the midpoint:
[tex]\[ \text{midpoint}_x = \frac{x_1 + x_2}{2} = \frac{14 + 10}{2} = \frac{24}{2} = 12 \][/tex]
3. Calculate the y-coordinate of the midpoint:
[tex]\[ \text{midpoint}_y = \frac{y_1 + y_2}{2} = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -1.5 \][/tex]
Therefore, the coordinates of the midpoint [tex]\(M\)[/tex] are:
[tex]\[ M = (12, -1.5) \][/tex]
Given the answer choices:
- A. [tex]\((6, -15)\)[/tex]
- B. [tex]\(\left(-2, -\frac{2}{2}\right)\)[/tex]
- C. [tex]\(\left(12, -\frac{3}{2}\right)\)[/tex]
- D. [tex]\((24, -3)\)[/tex]
- E. [tex]\((18, 12)\)[/tex]
The correct answer is:
C. [tex]\(\left(12, -\frac{3}{2}\right)\)[/tex]
This matches the coordinates [tex]\((12, -1.5)\)[/tex] obtained from our calculations.
