The area of the quadrilateral formed by the four lines 3x ± 4y ± 6 = 0 is equal to (in square units)

To find the area of the quadrilateral formed by the lines 3x ± 4y ± 6 = 0, we first need to identify the points of intersection of these lines. This involves solving the equations of the lines in pairs to find the vertices of the quadrilateral. Once we have those vertices, we can use the shoelace formula to calculate the area.

Identifying the Lines

The given equations can be interpreted as follows:

• 3x + 4y + 6 = 0
• 3x + 4y - 6 = 0
• 3x - 4y + 6 = 0
• 3x - 4y - 6 = 0

Finding Intersection Points

We will solve these equations pairwise to find the intersection points:

1. Intersection of 3x + 4y + 6 = 0 and 3x + 4y - 6 = 0

These lines are parallel and will not intersect.

2. Intersection of 3x + 4y + 6 = 0 and 3x - 4y + 6 = 0

To find the intersection, we can solve these equations simultaneously:

• From the first equation: 4y = -3x - 6 → y = (-3/4)x - 3/2
• Substituting into the second: 3x - 4((-3/4)x - 3/2) + 6 = 0

Simplifying this will give us the coordinates of one vertex.

3. Remaining Pairwise Intersections

By applying the same method to each pair of lines, we will find a total of four intersection points (vertices) for the quadrilateral.

Calculating the Area

Once we have the coordinates of the vertices, we can use the shoelace formula, which states:

If the vertices of the polygon are (x₁, y₁), (x₂, y₂), ..., (x_n, y_n), the area A is given by:

• A = 1/2 * |Σ (x_iy_i+1 - y_ix_i+1)| for i from 1 to n, where (x_n+1, y_n+1) = (x₁, y₁).

Example Calculation

Assume we find the vertices to be (a, b), (c, d), (e, f), and (g, h). Plugging these into the shoelace formula will yield the area. The specific values will depend on the intersections we calculated earlier.

Final Thoughts

By following these steps, you can systematically determine the area of the quadrilateral defined by the lines. The key is in accurately finding the intersection points and applying them to the shoelace formula correctly. If you have any specific intersection points you’ve calculated or need help with, feel free to share, and we can work through it together!