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 \(\left( \frac{7}{2}, 2 \right)\).
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