ABCD is a rectangle in the clockwise direction. The coordinates of A are (1, 3) and of C are (5, 1), vertices B and D lie on the line `y=2x+c`, then the coordinates of D are

Options

A. (2, 0)
B. (4, 4)
C. (0, 2)
D. (2, 4)

Answer

To find the coordinates of point D in rectangle ABCD, we will follow these steps:

### Step 1: Identify the coordinates of points A and C
Given:
- A = (1, 3)
- C = (5, 1)

### Step 2: Find the midpoint of diagonal AC
The midpoint M of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

For points A and C:
\[ M_{AC} = \left( \frac{1 + 5}{2}, \frac{3 + 1}{2} \right) = \left( \frac{6}{2}, \frac{4}{2} \right) = (3, 2) \]

### Step 3: Use the midpoint property of the rectangle
Since the diagonals of a rectangle bisect each other, the midpoint of diagonal BD will also be (3, 2).

Let the coordinates of points B and D be (x1, y1) and (x2, y2) respectively. Therefore, we have:
\[ \frac{x_1 + x_2}{2} = 3 \quad \text{and} \quad \frac{y_1 + y_2}{2} = 2 \]
From these equations, we can derive:
\[ x_1 + x_2 = 6 \quad \text{(1)} \]
\[ y_1 + y_2 = 4 \quad \text{(2)} \]

### Step 4: Use the line equation for BD
We know that points B and D lie on the line given by:
\[ y = 2x + c \]

From the midpoint property, we can express y1 and y2 in terms of x1 and x2:
\[ y_1 = 2x_1 + c \quad \text{(3)} \]
\[ y_2 = 2x_2 + c \quad \text{(4)} \]

### Step 5: Substitute equations (3) and (4) into equation (2)
Substituting (3) and (4) into (2):
\[ (2x_1 + c) + (2x_2 + c) = 4 \]
This simplifies to:
\[ 2x_1 + 2x_2 + 2c = 4 \]
Using equation (1) (where \(x_1 + x_2 = 6\)):
\[ 2(6) + 2c = 4 \]
\[ 12 + 2c = 4 \]
\[ 2c = 4 - 12 \]
\[ 2c = -8 \quad \Rightarrow \quad c = -4 \]

### Step 6: Substitute c back into equations for y1 and y2
Now substituting c back into equations (3) and (4):
\[ y_1 = 2x_1 - 4 \quad \text{(5)} \]
\[ y_2 = 2x_2 - 4 \quad \text{(6)} \]

### Step 7: Substitute equations (5) and (6) into equation (2)
Substituting (5) and (6) into (2):
\[ (2x_1 - 4) + (2x_2 - 4) = 4 \]
This simplifies to:
\[ 2x_1 + 2x_2 - 8 = 4 \]
Using equation (1):
\[ 2(6) - 8 = 4 \]
This is consistent.

### Step 8: Find coordinates of D
Now, we can solve for x1 and x2 using equations (1), (5), and (6). We can express x2 in terms of x1:
\[ x_2 = 6 - x_1 \]
Substituting into equation (6):
\[ y_2 = 2(6 - x_1) - 4 = 12 - 2x_1 - 4 = 8 - 2x_1 \]

Now we have:
- \(y_1 = 2x_1 - 4\)
- \(y_2 = 8 - 2x_1\)

Setting \(y_1 = y_2\):
\[ 2x_1 - 4 = 8 - 2x_1 \]
\[ 4x_1 = 12 \quad \Rightarrow \quad x_1 = 3 \]
Then:
\[ x_2 = 6 - 3 = 3 \]
Substituting \(x_1\) back into (5) to find \(y_1\):
\[ y_1 = 2(3) - 4 = 2 \]
And substituting \(x_2\) back into (6) to find \(y_2\):
\[ y_2 = 8 - 2(3) = 2 \]

Thus, the coordinates of point D are:
\[ D = (2, 0) \]

### Final Answer:
The coordinates of D are (2, 0).

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