Question

The points \( (1, 3) \) and \( (5, 1) \) are two opposite vertices of a rectangle. The other two vertices lie on the line \[ y = 2x + c, \quad \text{where} \quad c \text{ is the constant}, \quad \text{then the co-ordinates of the other two vertices are} \]

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

Hint:

In geometry problems involving rectangles, use the property that the diagonals bisect each other at the mid-point to find the coordinates of unknown vertices.

Answer 1

The Correct Option is A

Step 1: Understanding the problem.
We are given that the points \( (1, 3) \) and \( (5, 1) \) are opposite vertices of a rectangle, and the other two vertices lie on the line \( y = 2x + c \), where \( c \) is a constant. We are asked to find the coordinates of the other two vertices.

Step 2: Equation of the line.
We know that the mid-point of the diagonals of a rectangle coincide. Thus, the mid-point of the diagonal joining \( (1, 3) \) and \( (5, 1) \) is:
\[ \left( \frac{1 + 5}{2}, \frac{3 + 1}{2} \right) = (3, 2). \]
This point is also the mid-point of the other diagonal, which lies on the line \( y = 2x + c \). Substituting \( x = 3 \) into the equation of the line: \[ y = 2(3) + c = 6 + c. \]
Thus, the y-coordinate of the mid-point is \( 2 \), so:
\[ 6 + c = 2 \quad \Rightarrow \quad c = -4. \]

Step 3: Equation of the line.
Thus, the equation of the line becomes:
\[ y = 2x - 4. \]

Step 4: Finding the other vertices.
The other two vertices of the rectangle lie on this line, and the mid-point is \( (3, 2) \). Let the other two vertices be \( (x_1, y_1) \) and \( (x_2, y_2) \). Using the mid-point formula:
\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = (3, 2). \]
So, \( x_1 + x_2 = 6 \) and \( y_1 + y_2 = 4 \).
Substitute \( y_1 = 2x_1 - 4 \) and \( y_2 = 2x_2 - 4 \) into \( y_1 + y_2 = 4 \):
\[ (2x_1 - 4) + (2x_2 - 4) = 4 \quad \Rightarrow \quad 2x_1 + 2x_2 - 8 = 4 \quad \Rightarrow \quad 2x_1 + 2x_2 = 12 \quad \Rightarrow \quad x_1 + x_2 = 6. \]
Thus, the points are \( (4, 4) \) and \( (2, 0) \).
Final Answer:
The coordinates of the other two vertices are:
\[ \boxed{(4, 4), (2, 0)}. \]