Question

$A(-1,2)$, $B(5,1)$, $C(6,5)$ are the vertices of a parallelogram $ABCD$. The equation of the diagonal through $B$ is

• A. $x + y - 6 = 0$
• B. $x + y - 5 = 0$
• C. $x - y - 4 = 0$
• D. $x - y + 4 = 0$

Hint:

In a parallelogram, diagonals bisect each other. Use midpoint of one diagonal to find the fourth vertex, then form the line equation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The diagonals of a parallelogram bisect each other. Use this to find vertex $D$, then the equation of diagonal $BD$.
Step 2: Detailed Explanation:
Midpoint of $AC = \left(\dfrac{-1+6}{2},\dfrac{2+5}{2}\right) = \left(\dfrac{5}{2},\dfrac{7}{2}\right)$.
This is also the midpoint of $BD$, so $D = (5-5,\,7-1) = (0,6)$.
Slope of $BD = \dfrac{6-1}{0-5} = -1$. Equation: $y-1 = -1(x-5) \Rightarrow x+y-6=0$.
Step 3: Final Answer:
The equation of diagonal $BD$ is $x + y - 6 = 0$.