To make something seem less serious or wrong by offering explanations or reasons that partially excuse it; to reduce the apparent severity of something.
From Latin extenuare (to make thin or weak), from ex- (out) + tenuis (thin). The meaning shifted from physically thinning something to weakening or reducing the seriousness of an offense or fault through explanation.
In legal language, 'extenuating circumstances' can literally reduce your sentence—the word came from an old legal tradition of 'weakening' a case through mitigating factors, connecting thinness to reduced guilt!
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