If the length of the sides of a right triangle are in
a.p. then cosines of their acute angles are..........
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Let's say that their sides are a-r, a, a+r (This sequence is an AS).
We have, (a-r)^2 + a^2 = (a+r)^2
:: a^2 - 2ar + r^2 + a^2 = a^2 + 2ar + r^2
:: a^2 = 4ar
a cannot be zero as it is the length of the second side.
:: a = 4r
Hence, the side lengths are 3r, 4r, 5r respectively.
Cosine of the angles are 4/5, 0, 3/5.
This does not seem to be in any type of known progression.
We have, (a-r)^2 + a^2 = (a+r)^2
:: a^2 - 2ar + r^2 + a^2 = a^2 + 2ar + r^2
:: a^2 = 4ar
a cannot be zero as it is the length of the second side.
:: a = 4r
Hence, the side lengths are 3r, 4r, 5r respectively.
Cosine of the angles are 4/5, 0, 3/5.
This does not seem to be in any type of known progression.
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