A (1, -5), B (2, 2) and C (-2, 4) are the vertices of triangle ABC. find the equation of:
the line through C and parallel to AB. 

Answer

To find the equation of the line through point C (-2, 4) that is parallel to line AB, we will follow these steps:

### Step 1: Identify the coordinates of points A and B
The coordinates of the points are:
- A(1, -5)
- B(2, 2)
- C(-2, 4)

### Step 2: Calculate the slope of line AB
The slope (m) of a line through two points (x1, y1) and (x2, y2) is given by the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]

For points A(1, -5) and B(2, 2):
- \( y1 = -5 \)
- \( y2 = 2 \)
- \( x1 = 1 \)
- \( x2 = 2 \)

Substituting these values into the slope formula:
\[ m_{AB} = \frac{2 - (-5)}{2 - 1} = \frac{2 + 5}{1} = \frac{7}{1} = 7 \]

### Step 3: Use the slope to find the equation of the line through point C
Since the line through C is parallel to AB, it will have the same slope, which is 7. We will use the point-slope form of the equation of a line, which is:
\[ y - y1 = m(x - x1) \]

Here, \( m = 7 \), \( x1 = -2 \), and \( y1 = 4 \):
\[ y - 4 = 7(x + 2) \]

### Step 4: Simplify the equation
Now, we will simplify the equation:
\[ y - 4 = 7x + 14 \]
\[ y = 7x + 14 + 4 \]
\[ y = 7x + 18 \]

### Step 5: Rearranging to standard form
To express the equation in standard form \( Ax + By + C = 0 \):
\[ 7x - y + 18 = 0 \]

### Final Answer
The equation of the line through point C and parallel to line AB is:
\[ 7x - y + 18 = 0 \]

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