Find the exact value of the hypotenuse of the triangle.

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This is a special triangle, called a 30˚-60˚-90˚ triangle in which the ratios of the sides are `1:sqrt(3):2.`

The 9 is the long leg as it is opposite the 60˚ angle.  Therefore, the ratio of the long leg to the hypotenuse must be `sqrt(3)/2.`

This means we can write the proportion:

`9/x =` `sqrt(3)/2` where x represents the length of the hypotenuse.

`18 =sqrt(3)x`

`x = 18/sqrt(3)`  Must rationalize the denominator.

`18/sqrt(3) *sqrt(3)/sqrt(3)` = `(18sqrt(3))/3 = 6sqrt(3)`

The length of the hypotenuse is `6sqrt(3).`

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This is a 30-60-90 right triangle, where the the relation of the sides are:

hypotenuse =x

side opposite to 30 degree angle = x/2

side opposite to 60 degree angle=`sqrt3/2` x

The side opposite to the 60 degrees angle is 9, then

9=`sqrt3/2` x


Thus the exact value of the hypotenuse's length is `18/sqrt3`

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