Given the quadrilateral ABCD with the following vertices A(-4, 0), B(10, 0), C(10, 10), and D(0, 14), find its area.
After you sketch it, it is easy to see that you can split the area up into 2 triangle and a square, whose areas are 28, 20 and 100 , for a total of 148
or
list the coordinates in a column, repeating the first point
-4 0
10 0
10 10
0 14
-4 0
the area is (1/2)( sum of downproducts - sum of upproducts)
- (1/2)(0+100+140+0 -(0+0+0-56)
= (1/2(240 + 56) = 148
To find the area of the quadrilateral ABCD, we can divide it into two triangles and then calculate the area of each triangle separately.
Step 1: Calculate the length of the base of Triangle ABC.
The base of Triangle ABC is the distance between points A and C. Using the distance formula, we have:
AC = √[(x2 - x1)^2 + (y2 - y1)^2]
AC = √[(10 - (-4))^2 + (10 - 0)^2]
AC = √[(10 + 4)^2 + 10^2]
AC = √[14^2 + 10^2]
AC = √(196 + 100)
AC = √296
AC ≈ 17.20
Step 2: Calculate the height of Triangle ABC.
The height of Triangle ABC is the distance between line segments AB and CD. Since AB and CD are parallel, the height can be found by calculating the perpendicular distance between them. Let's find the equation of line AB first.
The slope of AB = (y2 - y1) / (x2 - x1)
= (0 - 0) / (10 - (-4))
= 0 / 14
= 0
Since the slope is 0, the equation of line AB is y = 0.
Now, let's find the distance between line AB and point C.
Distance = | (Ax + By + C) / sqrt(A^2 + B^2) | --(formula for the distance between a point and a line)
= | (10*0 + 1*10 + 0) / sqrt(0^2 + 1^2) |
= 10 / 1
= 10
Therefore, the height of Triangle ABC is 10.
Step 3: Calculate the area of Triangle ABC.
The area of Triangle ABC can be found using the formula: Area = (base * height) / 2.
Area_ABC = (AC * height) / 2
= (17.20 * 10) / 2
= 172 / 2
= 86
Step 4: Calculate the area of Triangle ACD.
Triangle ACD has the same base as Triangle ABC (AC = 17.20), but the height is different. Let's calculate the height of Triangle ACD.
From the previous calculations, we know that the distance between line AB and point C is 10. To find the distance between line AB and point D, we can use the same formula as above.
Distance = | (Ax + By + C) / sqrt(A^2 + B^2) |
= | (10*14 + 1*0 + 0) / sqrt(0^2 + 1^2) |
= 140 / 1
= 140
Therefore, the height of Triangle ACD is 140.
Step 5: Calculate the area of Triangle ACD.
Area_ACD = (AC * height) / 2
= (17.20 * 140) / 2
= 2408 / 2
= 1204
Step 6: Calculate the total area of the quadrilateral ABCD.
The total area of the quadrilateral ABCD is the sum of the areas of Triangle ABC and Triangle ACD.
Total Area_ABCD = Area_ABC + Area_ACD
= 86 + 1204
= 1290
Therefore, the area of quadrilateral ABCD is 1290 square units.
To find the area of the quadrilateral ABCD, you can use the Shoelace Formula, also known as the Gauss Area Formula. This formula calculates the area of any polygon given the coordinates of its vertices.
Here's how you can use the Shoelace Formula to find the area of quadrilateral ABCD:
1. List the coordinates of the vertices of the quadrilateral in a clockwise or counterclockwise order. Let's use the counterclockwise order: A(-4, 0), B(10, 0), C(10, 10), and D(0, 14).
2. Write the coordinates in two rows, with the x-coordinates in the first row and the y-coordinates in the second row:
x: -4 10 10 0
y: 0 0 10 14
3. Append the first x-coordinate and y-coordinate to the end of their respective rows:
x: -4 10 10 0 -4
y: 0 0 10 14 0
4. Multiply each x-coordinate by the y-coordinate immediately to its right, and then multiply each y-coordinate by the x-coordinate immediately to its right. Add up all these products:
(-4 × 0) + (10 × 0) + (10 × 10) + (0 × 14) + (-4 × 14) = 0 + 0 + 100 + 0 - 56 = 44
5. Take the absolute value of this sum and divide it by 2 to get the area of the quadrilateral:
|44| ÷ 2 = 22
Therefore, the area of quadrilateral ABCD is 22 square units.