The problem provides two points, A(-3, 2) and (5, 7), which are joined by a line segment. We are asked to find: a) The midpoint of the line segment. b) The equation of the line segment.

1. Problem Description

The problem provides two points, A(-3, 2) and (5, 7), which are joined by a line segment. We are asked to find:

a) The midpoint of the line segment.

b) The equation of the line segment.

2. Solution Steps

a) Finding the midpoint:

The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by the midpoint formula:

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

In our case, $(x_1, y_1) = (-3, 2)$ and $(x_2, y_2) = (5, 7)$. Plugging these values into the midpoint formula, we have:

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

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

$M = (1, 4.5)$

b) Finding the equation of the line:

First, we need to find the slope (m) of the line using the formula:

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

Using the points A(-3, 2) and (5, 7):

$m = \frac{7 - 2}{5 - (-3)} = \frac{5}{8}$

Now we can use the point-slope form of a linear equation:

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

Using point A(-3, 2) and the slope $m = \frac{5}{8}$:

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

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

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

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

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

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

We can also express this in the form $Ax + By = C$. Multiplying by 8, we get:

$8y = 5x + 31$

$-5x + 8y = 31$

$5x - 8y = -31$

3. Final Answer

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

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