Calculate the distance between the points (3, 7) and (5, 8), and find their midpoint.

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Given points are $(x_1, y_1) = (3, 7)$ and $(x_2, y_2) = (5, 8)$.

Substitute the coordinates into the distance formula:

$$D = \sqrt{(5 - 3)^2 + (8 - 7)^2}$$ $$D = \sqrt{(2)^2 + (1)^2}$$ $$D = \sqrt{4 + 1}$$ $$D = \sqrt{5}$$

The distance between the points is $\sqrt{5}$.

The midpoint formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Given points are $(x_1, y_1) = (3, 7)$ and $(x_2, y_2) = (5, 8)$.

Substitute the coordinates into the midpoint formula:

$$M = \left(\frac{3 + 5}{2}, \frac{7 + 8}{2}\right)$$ $$M = \left(\frac{8}{2}, \frac{15}{2}\right)$$ $$M = \left(4, \frac{15}{2}\right)$$

The midpoint of the points is $\left(4, \frac{15}{2}\right)$.

a) The distance between the points is $\boxed{\sqrt{5}}$.

b) The mid-point of the points is $\boxed{\left(4, \frac{15}{2}\right)}$.