Question

In triangle ABC, DE is parallel to AB. If DE is one-fifth AB, what is the ratio of the area of triangle DEC to the area of triangle ABC?

Answer 1

for any similar figures the ratio of areas is the square of the ratio of lengths.
so
1/25

You can easily prove this for yourself by noting that if DE = AB/4
Then the altitude of the little triangle = 1/5 the altitude of the big triangle
(1/2)(1/5)(1/5) = (1/25) * (1/2) (1)(1)

Answer 2

by the way, for similar SOLID 3 d figures:
areas are proportional to square of length ratio
Volumes are proportional to CUBE of scale ratio
This is handy to use for quick estimates
For example compare a 500 foot long oil tanker to a similar 1000 foot tanker
the big one has 2^2 = 4 times the surface area in the water
however it is 2^3 = 8 times the volume and therefore cargo capacity.
The water resistance is roughly proportional to area
so it carries 8 times the oil with 4 times the drag
so it carries twice as much oil per horsepower for the same speed.
That is why ships keep getting bigger and bigger.

Answer 3

You can easily prove this for yourself by noting that if DE = AB/"5" **not four**