1. **State the problem:** We have two similar triangles $\triangle GHI \sim \triangle JKL$ with corresponding sides $GI = 2$, $HI = 6$, $GH = 5$ and $JL = 12$, $JK = 30$, $KL = y$. We need to find the value of $y$. 2. **Recall the property of similar triangles:** Corresponding sides of similar triangles are proportional. This means: $$\frac{GI}{JL} = \frac{HI}{JK} = \frac{GH}{KL}$$ 3. **Set up the proportion using known sides:** $$\frac{GI}{JL} = \frac{2}{12} = \frac{1}{6}$$ $$\frac{HI}{JK} = \frac{6}{30} = \frac{1}{5}$$ Since these two ratios are not equal, check the other pair: $$\frac{GH}{KL} = \frac{5}{y}$$ 4. **Identify the correct corresponding sides:** Given the similarity notation $G \leftrightarrow J$, $H \leftrightarrow K$, $I \leftrightarrow L$, the sides correspond as follows: - $GI$ corresponds to $JL$ - $HI$ corresponds to $KL$ - $GH$ corresponds to $JK$ So the correct proportions are: $$\frac{GI}{JL} = \frac{HI}{KL} = \frac{GH}{JK}$$ 5. **Use the correct proportion to solve for $y$:** $$\frac{HI}{KL} = \frac{GH}{JK}$$ Substitute known values: $$\frac{6}{y} = \frac{5}{30}$$ 6. **Cross multiply:** $$6 \times 30 = 5 \times y$$ $$180 = 5y$$ 7. **Divide both sides by 5:** $$\cancel{\frac{180}{5}} = \cancel{\frac{5y}{5}}$$ $$36 = y$$ **Final answer:** $$y = 36$$