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The problem you have provided asks for several calculations based on the points \( A(-4, -3) \) and \( B(5, -2) \). The tasks include:

1. Finding the equation of the line through points \( A \) and \( B \).
2. Calculating the distance between points \( A \) and \( B \).
3. Finding the midpoint of the segment connecting points \( A \) and \( B \).
4. Determining the angle of the line.
5. Plotting the graph of the line.

### Step 1: Equation of the Line
To find the equation of the line through points \( A(x_1, y_1) = (-4, -3) \) and \( B(x_2, y_2) = (5, -2) \), we use the slope formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-3)}{5 - (-4)} = \frac{-2 + 3}{5 + 4} = \frac{1}{9}
\]

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

\[
y - y_1 = m(x - x_1)
\]

Substituting \( m = \frac{1}{9} \) and point \( A(-4, -3) \):

\[
y + 3 = \frac{1}{9}(x + 4)
\]

Simplifying:

\[
y + 3 = \frac{1}{9}x + \frac{4}{9}
\]

\[
y = \frac{1}{9}x + \frac{4}{9} - 3
\]

\[
y = \frac{1}{9}x - \frac{23}{9}
\]

Thus, the equation of the line is:

\[
y = \frac{1}{9}x - \frac{23}{9}
\]

### Step 2: Distance Between Points
The distance between two points is given by the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates:

\[
d = \sqrt{(5 - (-4))^2 + (-2 - (-3))^2} = \sqrt{(5 + 4)^2 + (-2 + 3)^2}
\]

\[
d = \sqrt{9^2 + 1^2} = \sqrt{81 + 1} = \sqrt{82} \approx 9.06
\]

### Step 3: Midpoint
The midpoint \( M \) of the segment joining two points is calculated as:

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

Substituting the coordinates:

\[
M = \left( \frac{-4 + 5}{2}, \frac{-3 + (-2)}{2} \right) = \left( \frac{1}{2}, \frac{-5}{2} \right)
\]

Thus, the midpoint is \( M\left( \frac{1}{2}, \frac{-5}{2} \right) \).

### Step 4: Angle of the Line
The angle \( \theta \) that the line makes with the positive \( x \)-axis can be found using the slope:

\[
\theta = \tan^{-1}(m)
\]

\[
\theta = \tan^{-1}\left( \frac{1}{9} \right)
\]

Let me calculate this.The angle \( \theta \) that the line makes with the positive \( x \)-axis is approximately \( 6.34^\circ \).

### Step 5: Graphing the Line
The equation of the line \( y = \frac{1}{9}x - \frac{23}{9} \) can be graphed by plotting points and using its slope. The slope tells us that for every 9 units the line moves horizontally, it rises by 1 unit.

If you'd like, I can create a graph of the line for you.

---

Do you have any questions or want more details on any part of the solution? Here are some related questions for deeper understanding:

1. How would the equation change if the slope was negative?
2. What if the points \( A \) and \( B \) were in a 3D plane? How would we calculate the distance?
3. What is the significance of the midpoint in geometry?
4. How would the distance formula change if the points involved had fractional coordinates?
5. Could the angle ever exceed \( 90^\circ \) for a line's slope?

**Tip:** When finding the angle of a line, always check the sign of the slope to determine if the angle is positive or negative.

Con los siguientes dos puntos, hallar la ecuación, calcular: distancia, punto medio, ángulo y graficar. A(-4, -3) y B(5, -2)

Learn how to calculate the equation of the line, distance, midpoint, and angle between points A(-4, -3) and B(5, -2) in this detailed solution using formulas from coordinate geometry. This is ideal for Grades 9-11 students.

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