What is the area of the composite figure whose vertices have the following coordinates?
(−8, 3) , (−4, 4) , (−1, 1) , (−4, −2) , (−8, −2)
well, on the left you have a four sided figure with parallel vertical left and right sides. The height on the left is 5 and on the right it is 6 so the average height is 5.5 and the base is 4 so that left partt has area = 4 * 5.5 = 22
then on the right you have an isoseles triangle with vertical base = 6 and altitude = -1 - -4 = 3 so area = 18
18+22 = 40
thank you very much sir
This was very useful, thanks nathan
To find the area of a composite figure, you need to break it down into separate shapes and then find the areas of those shapes individually. In this case, the given vertices form a polygon with five sides. We can break it down into two shapes: a rectangle and a triangle.
Step 1: Divide the figure into two shapes
Looking at the given vertices, we can see that the figure has four sides that form a rectangle, and an additional vertex that forms a triangle. So, we can divide the figure into a rectangle and a triangle.
Step 2: Find the dimensions of the rectangle
The rectangle is formed by the sides (-8, 3) - (-4, 4) and (-4, -2) - (-8, -2). To find the length of the rectangle, we subtract the x-coordinates of the two vertices: -8 - (-4) = -4. Similarly, to find the width of the rectangle, we subtract the y-coordinates of the two vertices: 4 - 3 = 1.
Step 3: Calculate the area of the rectangle
To find the area of a rectangle, multiply its length and width together. In this case, the length is -4 and the width is 1, so the area of the rectangle is -4 * 1 = -4 square units.
Step 4: Find the base and height of the triangle
The remaining vertex, (-1, 1), forms a triangle with two adjacent sides of the rectangle. We need to find the base and height of the triangle to calculate its area.
The base of the triangle is the distance between the vertices (-4, 4) and (-1, 1). To find the distance between two points in the coordinate plane, we can use the distance formula. The formula is given by:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the distance formula, we find the base to be sqrt((-1 - (-4))^2 + (1 - 4)^2) = sqrt(3^2 + (-3)^2) = sqrt(9 + 9) = sqrt(18) = 3sqrt(2).
The height of the triangle is the distance between the vertex (-1, 1) and the side of the rectangle formed by vertices (-4, -2) and (-8, -2). Since the y-coordinate of the triangle vertex is the same as the y-coordinate of the side of the rectangle, the height is the difference between the x-coordinates: -1 - (-8) = 7.
Step 5: Calculate the area of the triangle
To find the area of a triangle, multiply its base and height together and divide the product by 2. In this case, the base is 3sqrt(2) and the height is 7, so the area of the triangle is (3sqrt(2) * 7) / 2 = (21sqrt(2)) / 2 = 10.5sqrt(2) square units.
Step 6: Find the area of the composite figure
To find the total area of the composite figure, add the area of the rectangle and the area of the triangle together. In this case, the rectangle has an area of -4 square units and the triangle has an area of 10.5sqrt(2) square units. So the total area is -4 + 10.5sqrt(2) square units.
Therefore, the area of the composite figure is -4 + 10.5sqrt(2) square units.