1. **Problem statement:** Find the values of $x$, $y$, and $z$ in the right triangle with hypotenuse $14\sqrt{2}$ and angles $45^\circ$ and $30^\circ$. 2. **Relevant formulas and rules:** - In a right triangle, the sum of angles is $180^\circ$. - The side opposite $30^\circ$ is half the hypotenuse. - The side opposite $45^\circ$ in a $45^\circ-45^\circ-90^\circ$ triangle is $\frac{\text{hypotenuse}}{\sqrt{2}}$. - Use sine and cosine for non-special angles. 3. **Identify angles:** - Given angles are $45^\circ$ and $30^\circ$, so the right angle is $90^\circ$. 4. **Calculate sides:** - Hypotenuse $z = 14\sqrt{2}$ (given). - Side opposite $30^\circ$ is $x = \frac{z}{2} = \frac{14\sqrt{2}}{2} = 7\sqrt{2}$. 5. **Calculate side $y$ using Pythagoras:** $$y = \sqrt{z^2 - x^2} = \sqrt{(14\sqrt{2})^2 - (7\sqrt{2})^2} = \sqrt{392 - 98} = \sqrt{294} = 7\sqrt{6}$$ 6. **Final answers:** - $x = 7\sqrt{2}$ - $y = 7\sqrt{6}$ - $z = 14\sqrt{2}$