The points (1 , 3) and (5 , 1) are two opposite vertices of a rectangle . The other vertices lie on the line y = 2x + c , what is the value of c ?

Options

A. 2
B. `-2`
C. 4
D. `-4`

Answer

To find the value of \( c \) for the line \( y = 2x + c \) such that the other two vertices of the rectangle lie on this line, we can follow these steps:

### Step 1: Identify the given points
The two opposite vertices of the rectangle are given as \( A(1, 3) \) and \( B(5, 1) \).

### Step 2: Find the midpoint of the diagonal
The midpoint \( M \) of the diagonal \( AB \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \( A \) and \( B \):
\[ M = \left( \frac{1 + 5}{2}, \frac{3 + 1}{2} \right) = \left( \frac{6}{2}, \frac{4}{2} \right) = (3, 2) \]

### Step 3: Substitute the midpoint into the line equation
Now, we know that the point \( M(3, 2) \) lies on the line \( y = 2x + c \). We can substitute \( x = 3 \) and \( y = 2 \) into the equation to find \( c \):
\[ 2 = 2(3) + c \]

### Step 4: Solve for \( c \)
Now, simplify the equation:
\[ 2 = 6 + c \]
Subtract 6 from both sides:
\[ c = 2 - 6 \]
\[ c = -4 \]

### Conclusion
The value of \( c \) is \( -4 \).

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