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geometry

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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?

  • geometry -

    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)

  • geometry -

    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.

  • typo -

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

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