1. **State the problem:** We have two similar quadrilaterals FGHI and JKLM. We know sides FG = 45, HI = 40.2, and JK = 15. We need to find the length of side KL. 2. **Recall the property of similar figures:** Corresponding sides of similar figures are proportional. This means the ratio of any two corresponding sides in FGHI and JKLM is the same. 3. **Identify corresponding sides:** FG corresponds to JK, HI corresponds to LM, and GH corresponds to KL. We want to find KL. 4. **Set up the proportion using known sides:** $$\frac{FG}{JK} = \frac{GH}{KL}$$ We know $FG=45$, $JK=15$, and we want $KL$. We need $GH$ to use this proportion, but it is not given. However, since HI corresponds to LM, and HI = 40.2, we can find the scale factor first. 5. **Find the scale factor:** $$\text{scale factor} = \frac{JK}{FG} = \frac{15}{45} = \frac{1}{3}$$ 6. **Use the scale factor to find KL:** Since KL corresponds to GH, and the scale factor is $\frac{1}{3}$, then $$KL = GH \times \frac{1}{3}$$ But GH is not given, so we cannot find KL directly this way. 7. **Alternative approach:** Since HI corresponds to LM, and HI = 40.2, then $$LM = HI \times \frac{JK}{FG} = 40.2 \times \frac{15}{45} = 40.2 \times \frac{1}{3} = 13.4$$ 8. **Assuming GH corresponds to KL and GH equals HI (or similar), if GH is not given, we cannot find KL exactly without more information.** **Conclusion:** With the given data, the best we can do is find the scale factor $\frac{1}{3}$ and use it to find corresponding sides. If GH is known, then $$KL = GH \times \frac{1}{3}$$ Since the problem asks for KL and only JK is given on the smaller figure, the answer is $$KL = GH \times \frac{1}{3}$$ If GH is not given, the problem cannot be solved exactly. **Final answer:** $KL = GH \times \frac{1}{3}$ (needs GH to compute exact length).