The point of intersection of diagonals of a square ABCD is at the origin and one of its vertices is at a(4,2). What is the equation of the diagonal BD?

Options

A. `2x+y=0`
B. `2x-y=0`
C. `x+2y=0`
D. `x-2y=0`

Answer

To find the equation of diagonal BD of square ABCD, given that the point of intersection of the diagonals is at the origin (0, 0) and one vertex A is at (4, 2), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates of Vertex A**:
- We know that vertex A is at the coordinates (4, 2).

2. **Determine the Center of the Square**:
- The point of intersection of the diagonals is the center of the square, which is given as the origin (0, 0).

3. **Find the Coordinates of the Opposite Vertex C**:
- Since the diagonals of a square bisect each other at the center, the coordinates of the opposite vertex C can be found using the midpoint formula.
- If A is (4, 2) and C is (x, y), then:
\[ \left(\frac{4 + x}{2}, \frac{2 + y}{2}\right) = (0, 0) \]
- This gives us two equations:
\[ \frac{4 + x}{2} = 0 \quad \Rightarrow \quad 4 + x = 0 \quad \Rightarrow \quad x = -4 \]
\[ \frac{2 + y}{2} = 0 \quad \Rightarrow \quad 2 + y = 0 \quad \Rightarrow \quad y = -2 \]
- Therefore, the coordinates of vertex C are (-4, -2).

4. **Determine the Slope of Diagonal AC**:
- The slope (m) of line AC can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 2}{-4 - 4} = \frac{-4}{-8} = \frac{1}{2} \]

5. **Find the Slope of Diagonal BD**:
- Since the diagonals of a square are perpendicular, the slope of diagonal BD (let's call it m2) is the negative reciprocal of the slope of AC:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{1}{2}} = -2 \]

6. **Write the Equation of Diagonal BD**:
- Using the point-slope form of the equation of a line (y - y1 = m(x - x1)), where (x1, y1) is the origin (0, 0):
\[ y - 0 = -2(x - 0) \quad \Rightarrow \quad y = -2x \]
- Rearranging gives us the standard form:
\[ 2x + y = 0 \]

### Final Answer:
The equation of diagonal BD is:
\[ 2x + y = 0 \]