Question

13. Given the points (8, - 1) and (-1, 5), Find the exact distance between the points and the midpoint o the line segment whose endpoints are the given points.

Answer 1

To find the exact distance between the points \((8, -1)\) and \((-1, 5)\), we use the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Here, \((x_1, y_1) = (8, -1)\) and \((x_2, y_2) = (-1, 5)\). Substituting these values into the formula:

\[
d = \sqrt{((-1) - 8)^2 + (5 - (-1))^2}
\]

Calculating inside the square root:

\[
d = \sqrt{(-1 - 8)^2 + (5 + 1)^2}
\]

\[
d = \sqrt{(-9)^2 + (6)^2}
\]

\[
d = \sqrt{81 + 36}
\]

\[
d = \sqrt{117}
\]

Now, simplifying \(\sqrt{117}\):
\[
d = \sqrt{9 \cdot 13} = 3\sqrt{13}
\]

So, the exact distance between the points \((8, -1)\) and \((-1, 5)\) is \(3\sqrt{13}\).

Next, we find the midpoint of the line segment whose endpoints are the points \((8, -1)\) and \((-1, 5)\). The midpoint formula is given by:

\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Substituting the given values:

\[
\left( \frac{8 + (-1)}{2}, \frac{-1 + 5}{2} \right)
\]

Calculating each part:

\[
= \left( \frac{8 - 1}{2}, \frac{-1 + 5}{2} \right)
\]

\[
= \left( \frac{7}{2}, \frac{4}{2} \right)
\]

\[
= \left( \frac{7}{2}, 2 \right)
\]

Therefore, the midpoint of the line segment is \(\left( \frac{7}{2}, 2 \right)\).

In summary:
- The exact distance between the points is \(3\sqrt{13}\).
- The midpoint of the line segment is \(\left( \frac{7}{2}, 2 \right)\).