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# In a 30-60-90 triangle, where the long leg is 12, what is the length of the short leg?

Nov 15, 2015

The length of short leg$= 6.928$

#### Explanation:

${30}^{o}$-${60}^{o}$-${90}^{o}$ is a special kind of right-triangle in which sides exist in ratio $S L : L L : H = 1 : \sqrt{3} : 2$

where
$S L =$Short Leg,
$L L =$Long Leg,
$H =$Hypotenuse

The side-lengths can also be calculated with these relations
$S L = \frac{1}{2} H$ or $S L = \frac{1}{\sqrt{3}} L L$
$L L = \frac{\sqrt{3}}{2} H$ or $L L = S L \sqrt{3}$

Therefore, if $L L = 12$

$S L = \frac{1}{\sqrt{3}} \times 12 = 4 \sqrt{3} = 6.928$