If the middle points of the sides `B C ,C A ,` and `A B` of triangle `A B C` are `(1,3),(5,7),` and `(-5,7),` respectively, then find the equation of the side `A Bdot`

Explanation

To find the equation of the side \( AB \) of triangle \( ABC \) given the midpoints of its sides, we can follow these steps:

### Step 1: Identify the midpoints
The midpoints of the sides are given as:
- Midpoint of \( BC \): \( D(1, 3) \)
- Midpoint of \( CA \): \( E(5, 7) \)
- Midpoint of \( AB \): \( F(-5, 7) \)

### Step 2: Understand the relationship between the midpoints and the triangle
Since \( D \) and \( E \) are midpoints of sides \( BC \) and \( CA \) respectively, the line segment \( DE \) is parallel to side \( AB \). This means that the slope of line \( DE \) will be the same as the slope of line \( AB \).

### Step 3: Calculate the slope of line \( DE \)
The slope \( m \) of line \( DE \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points \( D(1, 3) \) and \( E(5, 7) \):
\[ m_{DE} = \frac{7 - 3}{5 - 1} = \frac{4}{4} = 1 \]

### Step 4: Use the slope to find the equation of line \( AB \)
Since \( AB \) is parallel to \( DE \), the slope of line \( AB \) is also \( 1 \). We can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Here, we can use the midpoint \( F(-5, 7) \) which lies on line \( AB \) and the slope \( m = 1 \):
\[ y - 7 = 1(x + 5) \]

### Step 5: Simplify the equation
Now, we can simplify the equation:
\[ y - 7 = x + 5 \]
\[ y = x + 12 \]

### Final Equation
Thus, the equation of the line \( AB \) is:
\[ y = x + 12 \]