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Is it possible for the lengths of a right triangle to form a geometric sequence? If yes, determine the lengths and the measures of the acute angles. if it's not possible, explain why not.

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asked by sophie
  1. lengths of a right triangle? You mean the three sides?

    Try it: lengths a, ar, ar^2

    now see if Pythagoras postulate will work:

    c^2=a^2 + b^2

    Either a, or ar^2 is c. Try a first

    a^2=(ar)^2+(ar^2)^2
    dividing by a^2
    1=r^2 + r^4
    r^4+r^2-1=0

    r^2 = (1+-sqrt(5))/2
    r= 1.272
    so sides are a; 1.272a; 1.618a

    which means a was not the longest side.

    check:
    (1.618^2)=1^2+1.272^2

    acute angles?
    sinA= 1.272/1.618 Then B: sinB=1/1.618

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