Sure, let's solve the given problem step-by-step.
### Given Points:
[tex]\( A(-1, 5) \)[/tex] and [tex]\( B(4, -3) \)[/tex]
### 1. Distance Between Two Points
To find the distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
d = \sqrt{(4 - (-1))^2 + (-3 - 5)^2}
\][/tex]
[tex]\[
d = \sqrt{(4 + 1)^2 + (-3 - 5)^2}
\][/tex]
[tex]\[
d = \sqrt{5^2 + (-8)^2}
\][/tex]
[tex]\[
d = \sqrt{25 + 64}
\][/tex]
[tex]\[
d = \sqrt{89}
\][/tex]
[tex]\[
d \approx 9.434
\][/tex]
So, the distance between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is approximately [tex]\( 9.434 \)[/tex].
### 2. Midpoint of the Line Segment
The midpoint [tex]\( M \)[/tex] of a line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
M = \left( \frac{-1 + 4}{2}, \frac{5 + (-3)}{2} \right)
\][/tex]
[tex]\[
M = \left( \frac{3}{2}, \frac{2}{2} \right)
\][/tex]
[tex]\[
M = \left( 1.5, 1.0 \right)
\][/tex]
So, the midpoint of the line segment is [tex]\( (1.5, 1.0) \)[/tex].
### 3. Slope of the Line Segment
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
m = \frac{-3 - 5}{4 - (-1)}
\][/tex]
[tex]\[
m = \frac{-3 - 5}{4 + 1}
\][/tex]
[tex]\[
m = \frac{-8}{5}
\][/tex]
[tex]\[
m = -1.6
\][/tex]
So, the slope of the line segment joining points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( -1.6 \)[/tex].
### Summary:
- The distance between points [tex]\( A(-1, 5) \)[/tex] and [tex]\( B(4, -3) \)[/tex] is approximately [tex]\( 9.434 \)[/tex].
- The midpoint of the line segment joining these points is [tex]\( (1.5, 1.0) \)[/tex].
- The slope of the line segment is [tex]\( -1.6 \)[/tex].
