Question

For the points (-8, -1) and (2, -5),
(a) Find the exact distance between the points.
(b) Find the midpoint of the line segment whose endpoints are the given points.

Answer 1

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 \( \boxed{(-3, -3)} \).