Find the area of the quadrilateral whose vertices are at(1, 3), (-1,-6), (4, -3) & (6, 1).
list the ordered pairs in a column with the points going counter-clockwise,
starting at any point. Whatever point you start with, repeat it at the end
1 3
-1 -6
4 -3
6 1
1 3
area = sum of the product of the downwards diagonals - sum of the product of the upward diagonals
= (1/2) [(-1)(-3) + 4(1) + 6(3) - (4(-6) + 6(-3) + 1(1) ) ]
= (1/2)[3 + 4 + 18 - (-24 - 18 + 1)]
= (1/2)(25 - (-41))
= 33
check my arithmetic
To find the area of a quadrilateral whose vertices are given, we can use the Shoelace formula.
The Shoelace formula states that the area of a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) is given by:
Area = 1/2 * |(x1 * y2 + x2 * y3 + ... + xn * y1) - (y1 * x2 + y2 * x3 + ... + yn * x1)|
Let's calculate the area using the given coordinates:
Vertices: (1, 3), (-1, -6), (4, -3), (6, 1)
Area = 1/2 * |(1 * (-6) + (-1) * (-3) + 4 * 1 + 6 * 3) - (3 * (-1) + (-6) * 4 + (-3) * 6 + 1 * 1)|
Calculating the expression inside the absolute value:
Area = 1/2 * |(-6 + 3 + 4 + 18) - (-3 - 24 - 18 + 1)|
Simplifying further:
Area = 1/2 * |19 - (-44)| = 1/2 * |19 + 44|
Calculating the absolute value:
Area = 1/2 * |63|
Finally, calculating the area:
Area = 1/2 * 63 = 31.5 square units
Therefore, the area of the quadrilateral is 31.5 square units.
To find the area of a quadrilateral with given vertices, you can use the Shoelace Formula.
The Shoelace Formula can be used to calculate the area of any polygon on a coordinate plane. To apply this formula, follow these steps:
Step 1: Write down the coordinates of the vertices of the quadrilateral in clockwise or counterclockwise order. Let's label these points as A, B, C, and D.
A(1, 3)
B(-1, -6)
C(4, -3)
D(6, 1)
Step 2: Write the x-coordinates and the y-coordinates in separate columns, duplicating the first coordinate as the last coordinate to close the shape.
X-coordinates: 1, -1, 4, 6, 1
Y-coordinates: 3, -6, -3, 1, 3
Step 3: Multiply the corresponding x-coordinate with the next y-coordinate and the next x-coordinate with the corresponding y-coordinate.
X-coordinates: 1, -1, 4, 6, 1
Y-coordinates: 3, -6, -3, 1, 3
Multiply: (1 * -6) + (-1 * -3) + (4 * 1) + (6 * 3) + (1 * -3) = -6 + 3 + 4 + 18 - 3 = 16
Step 4: Multiply the corresponding y-coordinate with the next x-coordinate and the next y-coordinate with the corresponding x-coordinate.
X-coordinates: 1, -1, 4, 6, 1
Y-coordinates: 3, -6, -3, 1, 3
Multiply: (3 * -1) + (-6 * 4) + (-3 * 6) + (1 * 1) + (3 * 1) = -3 - 24 - 18 + 1 + 3 = -41
Step 5: Take the absolute value of the difference between the two results above and divide it by 2.
Area = |16 - (-41)| / 2 = |16 + 41| / 2 = 57 / 2 = 28.5
Therefore, the area of the given quadrilateral is 28.5 square units.