Finding the Equation of a Square's Diagonal BD

This problem involves using the properties of a square's diagonals and coordinate geometry to find the equation of a line.

Understanding Square Diagonals

A square has special properties related to its diagonals. These properties are key to solving this problem:

• The diagonals of a square bisect each other. This means they cut each other in half at their intersection point.
• The diagonals of a square intersect at the center of the square.
• The diagonals of a square are perpendicular to each other. This means they meet at a 90-degree angle.
• The diagonals of a square have equal length.

Applying the Given Information

We are given the following:

• The square is ABCD.
• The point where the diagonals intersect is the origin (0, 0). Let's call this point O.
• One vertex is A(4, 2).

Since the origin O(0, 0) is the intersection point of the diagonals and diagonals bisect each other, O is the midpoint of both diagonal AC and diagonal BD.

We are interested in the equation of diagonal BD.

Finding the Slope of Diagonal AC

Diagonal AC passes through vertex A(4, 2) and the center O(0, 0). We can find the slope of AC using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Using A(4, 2) as \((x_1, y_1)\) and O(0, 0) as \((x_2, y_2)\):

\(m_{AC} = \frac{0 - 2}{0 - 4} = \frac{-2}{-4} = \frac{1}{2}\)

So, the slope of diagonal AC is \(\frac{1}{2}\).

Finding the Slope of Diagonal BD

We know that the diagonals of a square are perpendicular to each other. This means diagonal BD is perpendicular to diagonal AC.

If two non-vertical lines are perpendicular, the product of their slopes is -1. Let \(m_{BD}\) be the slope of diagonal BD.

\(m_{BD} \times m_{AC} = -1\)

\(m_{BD} \times \frac{1}{2} = -1\)

To find \(m_{BD}\), we multiply both sides by 2:

\(m_{BD} = -1 \times 2 = -2\)

The slope of diagonal BD is -2.

Finding the Equation of Diagonal BD

We know that diagonal BD passes through the origin O(0, 0) and has a slope \(m_{BD} = -2\). We can use the point-slope form of the equation of a line, which is \(y - y_1 = m(x - x_1)\).

Using the point O(0, 0) as \((x_1, y_1)\) and \(m = -2\):

\(y - 0 = -2(x - 0)\)

\(y = -2x\)

To write this equation in the standard form \(Ax + By + C = 0\), we can move the term \(-2x\) to the left side:

\(2x + y = 0\)

This is the equation of the diagonal BD.

Conclusion

Based on the properties of a square and the given coordinates, the equation of the diagonal BD is \(2x + y = 0\).

Revision Table: Square Diagonals and Equations

Additional Information: Square Properties and Coordinates

Understanding the properties of geometric shapes in the coordinate plane is part of analytical geometry. For a square centered at the origin, if one vertex is (a, b), the other vertices can be found using rotations or midpoint formulas.

• Finding Vertex C: Since O(0, 0) is the midpoint of AC and A is (4, 2), the coordinates of C are (-4, -2). This is because the origin is the midpoint of (4, 2) and (-4, -2): \(\left(\frac{4+(-4)}{2}, \frac{2+(-2)}{2}\right) = \left(\frac{0}{2}, \frac{0}{2}\right) = (0, 0)\).
• Finding Vertices B and D: Vertex B and D are rotated 90 degrees from A and C respectively, relative to the center O. A 90-degree clockwise rotation of (x, y) around the origin is (y, -x). A 90-degree counter-clockwise rotation is (-y, x).
  • Rotating A(4, 2) by 90 degrees (either way, BD passes through these): A 90-degree counter-clockwise rotation of A(4, 2) is (-2, 4). Let's say B is (-2, 4).
  • The other vertex D must be the opposite of B relative to the origin, so D would be (2, -4).
  • We can check if B(-2, 4), O(0, 0), and D(2, -4) are collinear and if the slope of BD is -2. The slope using B(-2, 4) and O(0, 0) is \(\frac{0-4}{0-(-2)} = \frac{-4}{2} = -2\). This matches our calculated slope for BD.
• Length of Diagonals: The length of diagonal AC can be found using the distance formula between A(4, 2) and C(-4, -2). Length of AC = \(\sqrt{(-4-4)^2 + (-2-2)^2} = \sqrt{(-8)^2 + (-4)^2} = \sqrt{64 + 16} = \sqrt{80}\). The length of the diagonal BD would also be \(\sqrt{80}\).