Find the exact value of the hypotenuse of the triangle.
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.
`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).`
This is a 30-60-90 right triangle, where the the relation of the sides are:
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
Thus the exact value of the hypotenuse's length is `18/sqrt3`