Question

Two sides of a parallelogram are along the lines $x+y=5$ and $x-y=-5.$ If the diagonals of the parallelogram intersect at (3, 6) then one of its vertices is at

• A. (6, 5)
• B. (7, 6)
• C. (7, 5)
• D. (6, 7)
• E. (5, 7)

Hint:

The point of intersection of diagonals in a parallelogram is the midpoint of each diagonal.

Answer 1

The Correct Option is D

Step 1: Analysis
One vertex (say A) is the intersection of $x+y=5$ and $x-y=-5$. Adding them: $2x=0 \implies x=0, y=5$. So $A = (0, 5)$.

Step 2: Analysis
The diagonals intersect at $M(3, 6)$. In a parallelogram, $M$ is the midpoint of the diagonal $AC$. Let $C = (x, y)$. Then $\frac{0+x}{2}=3$ and $\frac{5+y}{2}=6$.

Step 3: Calculation
$x=6$ and $y=7$. So the opposite vertex is $(6, 7)$. Final Answer: (D)