Let's solve the problem step by step.
### Part (a): Finding the Exact Distance Between the Points
We are given two points [tex]\((-2, 7)\)[/tex] and [tex]\((-6, 3)\)[/tex]. To find the distance between these two points, we use the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the given coordinates into the formula:
- [tex]\(x_1 = -2\)[/tex], [tex]\(y_1 = 7\)[/tex]
- [tex]\(x_2 = -6\)[/tex], [tex]\(y_2 = 3\)[/tex]
So, the distance is:
[tex]\[ \text{Distance} = \sqrt{((-6) - (-2))^2 + (3 - 7)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{(-6 + 2)^2 + (3 - 7)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{(-4)^2 + (-4)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{16 + 16} \][/tex]
[tex]\[ \text{Distance} = \sqrt{32} \][/tex]
[tex]\[ \text{Distance} = \sqrt{16 \times 2} \][/tex]
[tex]\[ \text{Distance} = 4\sqrt{2} \][/tex]
Converting [tex]\(\sqrt{32}\)[/tex] to its decimal form, we get:
[tex]\[ \text{Distance} \approx 5.656854249492381 \][/tex]
Thus, the exact distance between the points is [tex]\( \mathbf{4\sqrt{2}} \)[/tex] and its decimal approximation is [tex]\( \mathbf{5.656854249492381} \)[/tex].
### Part (b): Finding the Midpoint of the Line Segment
To find the midpoint of the line segment with endpoints [tex]\((-2, 7)\)[/tex] and [tex]\((-6, 3)\)[/tex], we use the midpoint formula:
[tex]\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Using the given coordinates:
- [tex]\(x_1 = -2\)[/tex], [tex]\(y_1 = 7\)[/tex]
- [tex]\(x_2 = -6\)[/tex], [tex]\(y_2 = 3\)[/tex]
So, the midpoint is:
[tex]\[ \text{Midpoint} = \left( \frac{-2 + (-6)}{2}, \frac{7 + 3}{2} \right) \][/tex]
[tex]\[ \text{Midpoint} = \left( \frac{-8}{2}, \frac{10}{2} \right) \][/tex]
[tex]\[ \text{Midpoint} = \left( -4, 5 \right) \][/tex]
Thus, the midpoint of the line segment is [tex]\( \mathbf{(-4, 5)} \)[/tex].
