Find the area of the trapezoid below by decomposing the shape into rectangles and triangles.
In order to find the area of the trapezoid, we need to decompose it into rectangles and triangles.
Let's label the trapezoid as ABCD, where AB is the top base and CD is the bottom base. Let's also label the intersection point of the diagonals as E.
To decompose the trapezoid, we can draw a line segment from point E perpendicular to AB and another line segment from point E perpendicular to CD. This creates two right triangles, AED and CED.
Next, we can see that the trapezoid is divided into two smaller triangles, AED and CED, and one rectangle, ABED.
Since we know that the area of a triangle is given by the formula (base * height) / 2 and the area of a rectangle is given by the formula length * width, we can calculate the area of each component and add them together to find the total area of the trapezoid.
First, let's label the given measurements:
AB = 8 units (length of the top base)
CD = 12 units (length of the bottom base)
Height of the trapezoid = 6 units (distance between AB and CD)
Height of triangle AED = 4 units (length of the line segment from point E perpendicular to AB)
Height of triangle CED = 2 units (length of the line segment from point E perpendicular to CD)
Now, let's calculate the area of each component:
Area of triangle AED = (AD * height of triangle AED) / 2 = (AB * height of triangle AED) / 2 = (8 * 4) / 2 = 16 square units
Area of triangle CED = (CD * height of triangle CED) / 2 = (12 * 2) / 2 = 12 square units
Area of rectangle ABED = AB * height of the trapezoid = 8 * 6 = 48 square units
Finally, let's add the areas of the components to find the total area of the trapezoid:
Total area = Area of triangle AED + Area of triangle CED + Area of rectangle ABED
Total area = 16 + 12 + 48
Total area = 76 square units
Therefore, the area of the trapezoid is 76 square units.