A(-1, 3), B(4, 2) and C(3, -2) are the vertices of a triangle.
Find the equation of the line through G and parallel to AC.

Answer

To find the equation of the line through the centroid \( G \) of triangle \( ABC \) and parallel to line segment \( AC \), we will follow these steps:

### Step 1: Find the Centroid \( G \) of Triangle \( ABC \)

The coordinates of the centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) are given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

Substituting the coordinates of points \( A(-1, 3) \), \( B(4, 2) \), and \( C(3, -2) \):
\[ G\left(\frac{-1 + 4 + 3}{3}, \frac{3 + 2 - 2}{3}\right) = G\left(\frac{6}{3}, \frac{3}{3}\right) = G(2, 1) \]

### Step 2: Find the Slope of Line Segment \( AC \)

The slope \( m \) of line segment \( AC \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( A(-1, 3) \) and \( C(3, -2) \):
\[ m = \frac{-2 - 3}{3 - (-1)} = \frac{-5}{4} \]

### Step 3: Write the Equation of the Line through \( G \) Parallel to \( AC \)

Since the line we want to find is parallel to \( AC \), it will have the same slope \( m = -\frac{5}{4} \). Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting the coordinates of \( G(2, 1) \) and the slope \( m = -\frac{5}{4} \):
\[ y - 1 = -\frac{5}{4}(x - 2) \]

### Step 4: Simplify the Equation

Expanding the equation:
\[ y - 1 = -\frac{5}{4}x + \frac{10}{4} \]
\[ y - 1 = -\frac{5}{4}x + \frac{5}{2} \]
Adding 1 to both sides:
\[ y = -\frac{5}{4}x + \frac{5}{2} + 1 \]
Converting 1 to a fraction:
\[ y = -\frac{5}{4}x + \frac{5}{2} + \frac{2}{2} \]
\[ y = -\frac{5}{4}x + \frac{7}{2} \]

### Step 5: Write in Standard Form

To convert this to standard form \( Ax + By + C = 0 \):
\[ \frac{5}{4}x + y - \frac{7}{2} = 0 \]
Multiplying through by 4 to eliminate the fraction:
\[ 5x + 4y - 14 = 0 \]

### Final Answer

The equation of the line through \( G \) and parallel to \( AC \) is:
\[ 5x + 4y - 14 = 0 \]

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