Suppose you are given triangle ABC with a line segment DE connecting side AB and BC so that DE is parallel to AC. The lengths of the sides of DBE are one-third the lengths of the sides of triangle ABC. What is the are of the trapezoid ADEC if the area of ABC is 27 cm?
First, we need to determine the area of triangle DBE. Since the sides of DBE are one-third the lengths of the sides of ABC, the area of DBE is (1/3)^2 = 1/9 times the area of ABC. Therefore, the area of DBE is 27 cm * (1/9) = 3 cm.
Next, we look at the trapezoid ADEC. Since DE is parallel to AC, triangle ADE and triangle CDE are similar to triangle ABC. This means that the ratio of the areas of triangle ADE and triangle ABC is equal to the square of the ratio of the sides, which is (1/3)^2 = 1/9.
Therefore, the area of trapezoid ADEC is equal to the area of ABC minus the area of DBE.
Area of ADEC = Area of ABC - Area of DBE
Area of ADEC = 27 cm - 3 cm = 24 cm
Therefore, the area of the trapezoid ADEC is 24 cm.