To find the distance between the points \((-8, -1)\) and \( (2, -5)\), we can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, we have:
- \( (x_1, y_1) = (-8, -1) \)
- \( (x_2, y_2) = (2, -5) \)
Substituting in the coordinates into the formula:
\[ d = \sqrt{(2 - (-8))^2 + (-5 - (-1))^2} \] \[ d = \sqrt{(2 + 8)^2 + (-5 + 1)^2} \] \[ d = \sqrt{(10)^2 + (-4)^2} \] \[ d = \sqrt{100 + 16} \] \[ d = \sqrt{116} \] \[ d = \sqrt{4 \cdot 29} \] \[ d = 2\sqrt{29} \]
So the exact distance between the points is \( \boxed{2\sqrt{29}} \).
Next, to find the midpoint of the line segment whose endpoints are the given points, we can use the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Using the coordinates from above:
\[ \text{Midpoint} = \left( \frac{-8 + 2}{2}, \frac{-1 + (-5)}{2} \right) \]
Calculating the x-coordinate:
\[ \frac{-8 + 2}{2} = \frac{-6}{2} = -3 \]
Calculating the y-coordinate:
\[ \frac{-1 - 5}{2} = \frac{-6}{2} = -3 \]
Thus, the midpoint is:
\[ \text{Midpoint} = (-3, -3) \]
So the midpoint of the line segment is \( \boxed{(-3, -3)} \).
1 answer