To find the distance between the points \( (9,1) \) and \( (-3,6) \), we can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (9, 1) \) and \( (x_2, y_2) = (-3, 6) \).
Plugging in the values:
\[ d = \sqrt{((-3) - 9)^2 + (6 - 1)^2} \]
Calculating the differences:
\[ d = \sqrt{(-12)^2 + (5)^2} \] \[ d = \sqrt{144 + 25} \] \[ d = \sqrt{169} \] \[ d = 13 \]
So, the distance between the two points is 13.0 units.
Next, to find the midpoint of the segment connecting the points \( (9,1) \) and \( (-3,6) \), we use the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Plugging in the coordinates:
\[ \text{Midpoint} = \left( \frac{9 + (-3)}{2}, \frac{1 + 6}{2} \right) \] \[ \text{Midpoint} = \left( \frac{6}{2}, \frac{7}{2} \right) \] \[ \text{Midpoint} = (3, 3.5) \]
So, the midpoint of the segment is \( (3, 3.5) \).
To summarize:
- The distance is 13.0 units.
- The midpoint is \( (3, 3.5) \).