To find the distance between the points (9,1) and (-3,6), we can use the distance formula, which is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (9, 1) \) and \( (x_2, y_2) = (-3, 6) \).
Plugging in the values:
\[ d = \sqrt{((-3) - (9))^2 + (6 - 1)^2} \] \[ = \sqrt{(-12)^2 + (5)^2} \] \[ = \sqrt{144 + 25} \] \[ = \sqrt{169} \] \[ = 13 \]
So, the distance between the two points is 13.0 units.
Now, let's find the midpoint of the segment connecting the two points. The midpoint \( M \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Using the same points:
\[ M = \left( \frac{9 + (-3)}{2}, \frac{1 + 6}{2} \right) \] \[ = \left( \frac{6}{2}, \frac{7}{2} \right) \] \[ = (3, 3.5) \]
Thus, the midpoint of the segment is \( (3, 3.5) \).
To summarize:
- Distance: 13.0 units
- Midpoint: \( (3, 3.5) \)