We are given a 30-60-90 triangle with the side opposite the 60 degree angle having length $\sqrt{6}$. We want to find the length of the hypotenuse $x$.

1. Problem Description

We are given a 30-60-90 triangle with the side opposite the 60 degree angle having length $\sqrt{6}$. We want to find the length of the hypotenuse $x$.

2. Solution Steps

In a 30-60-90 triangle, the sides are in the ratio $1:\sqrt{3}:2$, where the side opposite the 30 degree angle is 1, the side opposite the 60 degree angle is $\sqrt{3}$, and the hypotenuse is

2. Let the side opposite the 30 degree angle be $a$. Then the side opposite the 60 degree angle is $a\sqrt{3}$, and the hypotenuse is $2a$.

We are given that the side opposite the 60 degree angle is $\sqrt{6}$. Therefore,

$a\sqrt{3} = \sqrt{6}$

$a = \frac{\sqrt{6}}{\sqrt{3}} = \sqrt{\frac{6}{3}} = \sqrt{2}$

The hypotenuse $x$ is $2a = 2\sqrt{2}$.

3. Final Answer

$x = 2\sqrt{2}$