The problem asks to find the midpoint and the equation of the line segment joining points A(-3, 2) and B(5, 7).

1. Problem Description

The problem asks to find the midpoint and the equation of the line segment joining points A(-3, 2) and B(5, 7).

2. Solution Steps

a) Finding the midpoint:

The midpoint $M$ of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Here, $(x_1, y_1) = (-3, 2)$ and $(x_2, y_2) = (5, 7)$.

Therefore, the midpoint is:

$M = (\frac{-3 + 5}{2}, \frac{2 + 7}{2})$

$M = (\frac{2}{2}, \frac{9}{2})$

$M = (1, 4.5)$

b) Finding the equation of the line segment:

First, we find the slope $m$ of the line using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

$m = \frac{7 - 2}{5 - (-3)}$

$m = \frac{5}{8}$

Now, we use the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

Using point A(-3, 2):

$y - 2 = \frac{5}{8}(x - (-3))$

$y - 2 = \frac{5}{8}(x + 3)$

$y - 2 = \frac{5}{8}x + \frac{15}{8}$

$y = \frac{5}{8}x + \frac{15}{8} + 2$

$y = \frac{5}{8}x + \frac{15}{8} + \frac{16}{8}$

$y = \frac{5}{8}x + \frac{31}{8}$

So, the equation of the line is $y = \frac{5}{8}x + \frac{31}{8}$. We can also write it in the form $8y = 5x + 31$, or $5x - 8y + 31 = 0$.

3. Final Answer

a) The midpoint of the line segment is (1, 4.5).

b) The equation of the line segment is $y = \frac{5}{8}x + \frac{31}{8}$.