Step 1: Understanding the Concept:
In a parallelogram, the diagonals bisect each other. This means the midpoint of diagonal AC is the same as the midpoint of diagonal BD. The diagonal through B is the line BD.
: Key Formula or Approach:
1. Midpoint Formula: \( (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}) \).
2. Equation of line through two points.
Step 2: Detailed Explanation:
The diagonal through B is BD. Since the diagonals bisect each other, the midpoint of BD is the same as the midpoint of AC.
Midpoint of AC \( (M) \):
\[ M = \left( \frac{-1 + 6}{2}, \frac{2 + 5}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) \]
The line BD passes through \( B(5, 1) \) and \( M(2.5, 3.5) \).
Slope of BD:
\[ m = \frac{3.5 - 1}{2.5 - 5} = \frac{2.5}{-2.5} = -1 \]
Equation of line BD:
\[ y - 1 = -1(x - 5) \]
\[ y - 1 = -x + 5 \implies x + y - 6 = 0 \].
Step 3: Final Answer:
The equation of the diagonal is \( x + y - 6 = 0 \).