Determining the Area of a Quadrilateral with Zero Coordinates
In this article, we explore the problem of determining the area of a quadrilateral with specific vertices. Specifically, we will show that the area of a quadrilateral with vertices at (a, 0), (-b, 0), (0, -b), and (0, b) is zero, using the Shoelace Formula.
A General Method for Calculating the Area of a Polygon
To calculate the area of any polygon given its vertices, we can use the Shoelace Formula (also known as Gauss's area formula or the surveyor's formula). The formula is:
$ A frac{1}{2} left(sum_{i1}^{n} x_i y_{i 1} - y_i x_{i 1} right) $
where the subscript $i 1$ wraps around to 1 for the last vertex. This ensures the polygon is closed.
Identifying the Vertices
The vertices of our quadrilateral are:
(a, 0) (-b, 0) (0, -b) (0, b) (a, 0) (repeating the first vertex to close the shape)Applying the Shoelace Formula
Let's assign the vertices as follows:
(x1, y1) (a, 0) (x2, y2) (-b, 0) (x3, y3) (0, -b) (x4, y4) (0, b) (x5, y5) (a, 0)Now, we plug these into the Shoelace Formula:
$ A frac{1}{2} left(a cdot 0 (-b) cdot (-b) 0 cdot b 0 cdot 0 - (0 cdot (-b) 0 cdot 0 (-b) cdot 0 b cdot a) right) $
Let's calculate the terms:
$a cdot 0 0$ $(-b) cdot (-b) b^2$ $0 cdot b 0$ $0 cdot 0 0$ $0 cdot (-b) 0$ $0 cdot 0 0$ $(-b) cdot 0 0$ $b cdot a ab$Substituting these into the formula:
$ A frac{1}{2} left(0 b^2 0 0 - (0 0 0 ab) right) frac{1}{2} left(b^2 - ab right) $
Simplifying further:
$ A frac{1}{2} (b^2 - ab) frac{1}{2} (b(b - a)) $
Interpreting the Result
The area of the quadrilateral is zero if and only if $b a$. In any other case, where $b eq a$, the area is positive.
Therefore, the area of the quadrilateral is zero if and only if the value of $a$ and $b$ are equal.
The Result
Final result: The area of the quadrilateral is zero if and only if $a b$.
Note: If the quadrilateral appears to form a triangle due to one of the vertices along the x-axis and the other forming a vertical line, the area calculation would be different. However, in our specific case, the quadrilateral's area is shown to be zero.