#7. Find the area of a polygon with the vertices of (-4,5), (-1,5), (4,-3), and (-4,-3).
Note that this is just a trapezoid with bases 3 and 8, and height 8. So, the area is
(3+8)/2 * 8 = 44
Why did the polygon go to therapy?
Because it had too many unresolved vertices issues!
Now, let's find the area of this hilarious polygon. To do that, we can use the Shoelace Formula.
First, let's label the vertices A, B, C, and D, respectively, as follows:
A(-4, 5), B(-1, 5), C(4, -3), and D(-4, -3).
Now, we'll use the Shoelace Formula:
Area = ½ * |(Ax * By + Bx * Cy + Cx * Dy + Dx * Ay) - (Ay * Bx + By * Cx + Cy * Dx + Dy * Ax)|
Substituting the values:
Area = ½ * |((-4 * 5) + (-1 * (-3)) + (4 * (-3)) + (-4 * 5)) - ((5 * (-1)) + (-3 * 4) + (-3 * (-4)) + (5 * (-4)))|
Calculating it, we get:
Area = ½ * |(-20 + 3 - 12 - 20) - (-5 - 12 + 12 - 20)|
Area = ½ * |-49 - (-25)|
Area = ½ * |-49 + 25|
Area = ½ * |-24|
Area = 12
So, the area of this polygon is 12 square units. Keep polygon-ing, my friend!
To find the area of a polygon with the given vertices, you can use the Shoelace Formula. Here are the step-by-step instructions:
1. Write down the coordinates of the vertices in order, either clockwise or counterclockwise. Let's use clockwise order: (-4,5), (-1,5), (4,-3), and (-4,-3).
2. Multiply each x-coordinate with the y-coordinate of the next vertex in the order and write the result in a column. For example:
-4 x 5 = -20
-1 x -3 = 3
4 x -3 = -12
-4 x 5 = -20
The first three products are positioned vertically in a column.
3. Multiply each y-coordinate with the x-coordinate of the next vertex in the order and write the result in another column. For example:
5 x -1 = -5
5 x 4 = 20
-3 x -4 = 12
-3 x -4 = 12
The last product is positioned vertically in a separate column.
4. Sum up the values in the first column and the values in the second column separately. For example:
Sum of the values in the first column: -20 + 3 - 12 - 20 = -49
Sum of the values in the second column: -5 + 20 + 12 + 12 = 39
5. Take the absolute value of each of the sums calculated in the previous step. For example:
Absolute value of the sum from the first column: |-49| = 49
Absolute value of the sum from the second column: |39| = 39
6. Divide the absolute value of the sum from the second column by 2 to get the area of the polygon. For example:
Area = 39 / 2 = 19.5
The area of the polygon with the given vertices (-4,5), (-1,5), (4,-3), and (-4,-3) is 19.5 square units.
To find the area of a polygon, you can use the Shoelace Formula, also known as Gauss's area formula or the surveyor's formula. Here's how you can solve it step by step:
1. Write down the coordinates of the vertices in order, either clockwise or counterclockwise.
Given vertices: (-4,5), (-1,5), (4,-3), and (-4,-3).
Order of vertices: (-4,5), (-1,5), (4,-3), (-4,-3).
2. Multiply the x-coordinate of each vertex by the y-coordinate of the vertex immediately after it.
(-4 * 5) + (-1 * -3) + (4 * -3) + (-4 * 5)
3. Multiply the y-coordinate of each vertex by the x-coordinate of the vertex immediately after it.
(5 * -1) + (5 * 4) + (-3 * -4) + (-3 * -4)
4. Take the absolute value of the difference between the two sums.
|((-4 * 5) + (-1 * -3) + (4 * -3) + (-4 * 5)) - ((5 * -1) + (5 * 4) + (-3 * -4) + (-3 * -4))|
5. Divide the absolute value by 2 to obtain the area of the polygon.
|((-4 * 5) + (-1 * -3) + (4 * -3) + (-4 * 5)) - ((5 * -1) + (5 * 4) + (-3 * -4) + (-3 * -4))| / 2
Evaluating this expression will give you the area of the polygon.