1. **State the problem:** We are given two right triangles sharing a common hypotenuse line. The smaller triangle has legs $x$ and $h$ and hypotenuse $y$. The larger triangle has legs 5 and 12 and hypotenuse 13. We need to find the value of the ratio $\frac{y}{x}$. 2. **Recall the Pythagorean theorem:** For a right triangle with legs $a$ and $b$ and hypotenuse $c$, the relation is $$a^2 + b^2 = c^2.$$ This applies to both triangles. 3. **Apply the theorem to the larger triangle:** Given legs 5 and 12, and hypotenuse 13, check: $$5^2 + 12^2 = 25 + 144 = 169 = 13^2,$$ which confirms the triangle is right-angled. 4. **Use similarity of triangles:** Since the two triangles share the hypotenuse line and are right triangles, they are similar. Therefore, corresponding sides are proportional: $$\frac{y}{13} = \frac{h}{12} = \frac{x}{5}.$$ 5. **Express $y$ in terms of $x$ using the ratio:** From similarity, $$\frac{y}{13} = \frac{x}{5} \implies y = \frac{13}{5} x.$$ 6. **Find the ratio $\frac{y}{x}$:** $$\frac{y}{x} = \frac{13}{5}.$$ **Final answer:** $$\boxed{\frac{y}{x} = \frac{13}{5}}.$$