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Proof without words of the AM–GM inequality: PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ. Visual proof that (x + y) 2 ≥ 4xy. Taking square roots and dividing by two gives the AM ...
511 Gm (3.4 au) – average diameter of Mira, a pulsating red giant and the progenitor of the Mira variables. It is an asymptotic giant branch star. [182] 570 Gm (3.8 au) – length of the tail of Comet Hyakutake measured by Ulysses; the actual value could be much higher; 590 Gm (3.9 au) – diameter of the Pistol Star, a blue hypergiant star [183]
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Metric units are units based on the metre, gram or second and decimal (power of ten) multiples or sub-multiples of these. According to Schadow and McDonald, [1] metric units, in general, are those units "defined 'in the spirit' of the metric system, that emerged in late 18th century France and was rapidly adopted by scientists and engineers.
The stone is not a measurement of weight used in the US. The US customary system uses the short hundredweight of 100 lb and short ton of 2,000 lb. Where these systems most notably differ is in their units of volume. An imperial fluid ounce of 28.4130625 ml is 3.924% smaller than the US fluid ounce (fl oz) of 29.5735295625 millilitres (ml).
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Suppose AC = x 1 and BC = x 2. Construct perpendiculars to [AB] at D and C respectively. Join [CE] and [DF] and further construct a perpendicular [CG] to [DF] at G. Then the length of GF can be calculated to be the harmonic mean, CF to be the geometric mean, DE to be the arithmetic mean, and CE to be the quadratic mean.