geometry
posted by Danyoline on .
In triangle ABC, DE is parallel to AB. If DE is onefifth AB, what is the ratio of the area of triangle DEC to the area of triangle ABC?

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