Question

In △ABC a line segment DE connects sides AB and BC so that DE ∥ AC . The lengths of the sides of △DBE are four times shorter than the lengths of the sides of △ABC. What is the area of △ABC if the area of a trapezoid ADEC is 30cm^2?

Answer 1

The area of △ABC is found by understanding that the areas of △DBE and trapezoid ADEC are proportional to the areas of △ABC and △BEA, respectively. With the given area of ADEC (30 cm2), we calculate the area of △ABC to be 32 cm2.

Step-by-step explanation:

The problem involves finding the area of △ABC given that the area of the trapezoid ADEC is 30 cm2 and DE is parallel to AC. Because the sides of △DBE are four times shorter than the sides of △ABC, DE is one fourth of AC. This also means that the altitude of △DBE is four times shorter than the altitude of △ABC, and consequently, the area of △DBE is 1/16 the area of △ABC. Similarly, ADEC is a trapezoid with the same altitude as △ABC and with the bases being AC and DE (1/4 of AC).

To find the area of △ABC, we can use the fact that the area of △ABE is the sum of the areas of △DBE and trapezoid ADEC. As the area of △DBE is 1/16 that of △ABC, and the area of trapezoid ADEC is given as 30 cm2, we can express the area of △ABC as 16 times the area of △DBE. Letting the area of △ABC be A, we then have A + 30 cm2 = 16A. Solving for A, we find that A = 30/15 = 2 cm2, which would be the area of △DBE. The area of △ABC is therefore 16 * 2 cm2 = 32 cm2.

Answer 2

32 cm^2

Step-by-step explanation:

The sides of ∆DBE or 1/4 of the sides of ∆ABC, so its area is (1/4)^2 = 1/16 of the area of ∆ABC. The trapezoid's area is then 1 -1/16 = 15/16 of the total area of ∆ABC. Said another way, the area of ∆ABC is 16/15 times the area of the trapezoid.

Area ∆ABC = (16/15)·(30 cm^2) = 32 cm^2