1. **Stating the problem:** We have a right triangle with vertices $x$, $y$, and $z$. The segment $xy$ is vertical with length 12, $yz$ is horizontal, and $xz$ is the hypotenuse. We want to find the lengths of $xy$ and $xz$. 2. **Given information:** - Length of $xy = 12$ - Right angle at vertex $y$ 3. **Formula used:** In a right triangle, by the Pythagorean theorem, $$xz^2 = xy^2 + yz^2$$ 4. **Explanation:** We know $xy = 12$, but $yz$ is not given. Without $yz$, we cannot find the exact length of $xz$. However, $xy$ is already given as 12. 5. **Answer:** - $xy = 12$ - $xz = \sqrt{12^2 + yz^2} = \sqrt{144 + yz^2}$ (depends on $yz$ length) Since $yz$ is not provided, $xz$ cannot be numerically determined. If you provide the length of $yz$, I can calculate $xz$ for you.