If x = 5 units, y = 7 units, and h = 6 units, find the area of the trapezoid shown above using decomposition.
Area of trapezoid = (x + y) * h / 2
Area of trapezoid = (5 + 7) * 6 / 2
Area of trapezoid = 12 * 6 / 2
Area of trapezoid = 72 / 2
Area of trapezoid = 36 units^2
To find the area of the trapezoid using decomposition, we can divide it into two simpler shapes: a rectangle and a triangle. Then, we can calculate the individual areas of these shapes and add them together to find the total area of the trapezoid.
First, let's identify the different parts of the trapezoid. The longer parallel sides are called bases, and the shorter sides are called legs. In this case, the bases are the sides with lengths x = 5 units and y = 7 units. The perpendicular distance between the bases is called the height, which is h = 6 units.
Now, we can calculate the area of the trapezoid using decomposition:
1. Calculate the area of the rectangle: The base of the rectangle is the length y = 7 units, and the height is the same as the trapezoid's height, h = 6 units. The area of the rectangle is given by the formula: Area_rectangle = base * height = y * h = 7 * 6 = 42 square units.
2. Calculate the area of the triangle: The base of the triangle is the difference between the lengths of the two bases, so Base_triangle = y - x = 7 - 5 = 2 units. The height of the triangle is the same as the trapezoid's height, h = 6 units. The area of the triangle is given by the formula: Area_triangle = (1/2) * base * height = (1/2) * (y - x) * h = (1/2) * 2 * 6 = 6 square units.
3. Add the areas of the rectangle and the triangle to find the total area of the trapezoid: Area_trapezoid = Area_rectangle + Area_triangle = 42 + 6 = 48 square units.
Therefore, the area of the trapezoid is 48 square units.
To find the area of the trapezoid using decomposition, we need to divide it into simpler shapes and calculate their areas individually.
Step 1: Divide the trapezoid into two triangles and a rectangle.
First, draw a line from the top left corner (A) to the bottom right corner (D). This creates two triangles. Let's label the new point of intersection as E.
Step 2: Calculate the area of the rectangle.
The rectangle can be defined as the base of the trapezoid, with length h, and the height is given by the length of the side AD, which is y. So, the area of the rectangle is given by:
Area of Rectangle = base x height = h x y
Step 3: Calculate the areas of the two triangles.
The first triangle (AEB) has a base length of x and a height length of h-y. Therefore, its area can be calculated as:
Area of Triangle 1 = 0.5 x base x height = 0.5 x x x (h-y)
Similarly, the second triangle (CED) has a base length of x and a height length of y. Its area can be calculated as:
Area of Triangle 2 = 0.5 x base x height = 0.5 x x x y
Step 4: Add up the areas of the rectangle and the two triangles.
Total Area of Trapezoid = Area of Rectangle + Area of Triangle 1 + Area of Triangle 2
Total Area of Trapezoid = (h x y) + (0.5 x x x (h-y)) + (0.5 x x x y)
To find the numerical value of the area, substitute the given values for x, y, and h into the equation and calculate the result.