1. **State the problem:** We have two similar right triangles $\triangle ABC \sim \triangle XYZ$. Given sides are $AC=16$, $CB=30$, $AB=34$ in $\triangle ABC$, and $YZ=15$, $ZX=y$, $XY=x$ in $\triangle XYZ$. We need to find the missing side $y$ in $\triangle XYZ$. 2. **Recall the properties of similar triangles:** Corresponding sides of similar triangles are proportional. That means: $$\frac{AC}{YZ} = \frac{CB}{ZX} = \frac{AB}{XY}$$ 3. **Identify corresponding sides:** Since the triangles are similar and the right angle is at $C$ in $\triangle ABC$ and at $Z$ in $\triangle XYZ$, the sides correspond as: - $AC \leftrightarrow YZ$ - $CB \leftrightarrow ZX$ - $AB \leftrightarrow XY$ 4. **Use the known sides to find the scale factor:** $$\frac{AC}{YZ} = \frac{16}{15}$$ 5. **Set up the proportion to find $y = ZX$:** $$\frac{CB}{ZX} = \frac{16}{15} \implies \frac{30}{y} = \frac{16}{15}$$ 6. **Solve for $y$:** $$30 \times 15 = 16 \times y$$ $$450 = 16y$$ $$y = \frac{450}{16}$$ 7. **Simplify the fraction:** $$y = \frac{\cancel{450}^{\times 1}}{\cancel{16}^{\times 1}} = 28.125$$ 8. **Final answer:** The missing side $y$ in $\triangle XYZ$ is $28.125$.