Question

The Points (1,3), (5,1) are Opposite vertices of a diagonal of a rectangle. If the other two vertices lie on the line y=2x+c, then one of the vertex on the other diagonal is? 
 

• A. (1,-2)
• B. (0,-4)
• C. (2,0)
• D. (3,2)

Hint:

Equation of Line Passing Through Rectangle's Diagonal Midpoint
When the diagonals of a rectangle bisect each other, their intersection point is the midpoint of both diagonals.

Steps:

1. Find the midpoint: The coordinates of the midpoint \((3, 2)\) are calculated using the midpoint formula.
2. Equation of the line: Since the midpoint \((3, 2)\) lies on the line, and it is given as \(y = 2x + c\), we substitute to find \(c\).
3. Conclusion: Therefore, the equation of the line passing through the midpoint \((3, 2)\) of the diagonals of the rectangle is \(y = 2x - 4\).

Thus, the line equation is \(y = 2x - 4\).

Answer 1

The Correct Option is C

The points (1, 3) and (5, 1) are opposite vertices of a diagonal of a rectangle. We are given that the other two vertices lie on the line \( y = 2x + c \), and we need to find one of the vertices on the other diagonal.

Step 1: Midpoint of the diagonal:
The midpoint of the diagonal connecting the points (1, 3) and (5, 1) can be found by averaging the coordinates of these two points. The midpoint formula is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates of the given points: \[ M = \left( \frac{1 + 5}{2}, \frac{3 + 1}{2} \right) = (3, 2) \] So, the midpoint of the diagonal is \( (3, 2) \).

Step 2: Midpoint of the other diagonal:
The opposite diagonal of the rectangle will also have its midpoint at the same point \( (3, 2) \), as the diagonals of a rectangle bisect each other. If the other two vertices of the rectangle are \( (x_1, y_1) \) and \( (x_2, y_2) \), then the midpoint of this diagonal is: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = (3, 2) \] Therefore, we have the following system of equations: \[ \frac{x_1 + x_2}{2} = 3 \quad \text{and} \quad \frac{y_1 + y_2}{2} = 2 \] This simplifies to: \[ x_1 + x_2 = 6 \quad \text{and} \quad y_1 + y_2 = 4 \]

Step 3: Use the line equation:
We are told that the other two vertices lie on the line \( y = 2x + c \). Let’s substitute the coordinates of the points \( (x_1, y_1) \) and \( (x_2, y_2) \) into the equation of the line: \[ y_1 = 2x_1 + c \quad \text{and} \quad y_2 = 2x_2 + c \] Now substitute these into the equation for \( y_1 + y_2 = 4 \): \[ (2x_1 + c) + (2x_2 + c) = 4 \] Simplifying: \[ 2x_1 + 2x_2 + 2c = 4 \] Using the equation \( x_1 + x_2 = 6 \) from earlier: \[ 2(6) + 2c = 4 \quad \Rightarrow \quad 12 + 2c = 4 \quad \Rightarrow \quad 2c = -8 \quad \Rightarrow \quad c = -4 \] Therefore, the equation of the line becomes \( y = 2x - 4 \).

Step 4: Find the points on the line:
Now that we know the equation of the line is \( y = 2x - 4 \), we can substitute the values of \( x_1 \) and \( x_2 \) into this equation. We know that \( x_1 + x_2 = 6 \). Let’s solve for the coordinates of the points. Assume \( x_1 = 2 \) and \( x_2 = 4 \) (since their sum must be 6). Substituting into the line equation: \[ y_1 = 2(2) - 4 = 0 \quad \text{and} \quad y_2 = 2(4) - 4 = 4 \] So, the points are \( (2, 0) \) and \( (4, 4) \).

Final Answer:
One of the vertices on the other diagonal of the rectangle is \( \boxed{(2, 0)} \). Therefore, the correct option is (C) : (2, 0).