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: Understanding the Concept:
We are given two adjacent sides of a parallelogram and the point of intersection of its diagonals. The key properties are that vertices are intersections of sides, and the intersection of diagonals is the midpoint of each diagonal.
Step 2: Key Formula or Approach:
1. Solve a system of linear equations to find the intersection point (vertex) of the two given lines.
2. Use the midpoint formula: If M is the midpoint of a line segment AC, then \( M = \left(\frac{x_A+x_C}{2}, \frac{y_A+y_C}{2}\right) \).
Step 3: Detailed Explanation:
Step 3a: Find one vertex
The two given lines represent adjacent sides. Let's find their point of intersection, which will be one of the vertices of the parallelogram. Let's call this vertex A.
Equation 1: \( x + y = 5 \)
Equation 2: \( x - y = -5 \)
Adding the two equations:
\[ (x+y) + (x-y) = 5 + (-5) \] \[ 2x = 0 \implies x = 0 \] Substitute \( x=0 \) into Equation 1:
\[ 0 + y = 5 \implies y = 5 \] So, one vertex is \( A = (0, 5) \).
Step 3b: Find the opposite vertex
The intersection of the diagonals is the point \( M = (3, 6) \). This point is the midpoint of the diagonal connecting vertex A to its opposite vertex, say C.
Let the coordinates of vertex C be \( (x_C, y_C) \). Using the midpoint formula:
For the x-coordinate:
\[ 3 = \frac{0 + x_C}{2} \implies 6 = x_C \] For the y-coordinate:
\[ 6 = \frac{5 + y_C}{2} \implies 12 = 5 + y_C \implies y_C = 7 \] So, the vertex opposite to A is \( C = (6, 7) \).
This point \( (6,7) \) is one of the vertices of the parallelogram and it is listed as an option. We don't need to find the other two vertices.
Step 4: Final Answer:
One of the vertices of the parallelogram is (6, 7).