find the length of side $x$ to the nearest tenth.

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$, where the legs are of equal length and the hypotenuse $c$ is related to the leg length $a$ (or $b$) by $c = a\sqrt{2}$.

Step2: Set up the equation

Let the length of the leg be $x$. The hypotenuse is $\sqrt{6}$. Using the 45 - 45 - 90 triangle ratio $c=a\sqrt{2}$, we have $\sqrt{6}=x\sqrt{2}$.

Step3: Solve for $x$

To solve for $x$, divide both sides of the equation $\sqrt{6}=x\sqrt{2}$ by $\sqrt{2}$. So $x=\frac{\sqrt{6}}{\sqrt{2}}$. Rationalize the denominator: $x = \frac{\sqrt{6}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{12}}{2}=\frac{2\sqrt{3}}{2}=\sqrt{3}\approx1.7$.

Answer:

$1.7$