1. **State the problem:** We have two similar pentagons, A (smaller) and B (larger). We know the side lengths of pentagon A: 10 cm, 12 cm, 15 cm, 6 cm, and 9 cm. For pentagon B, three sides are given: $x$ cm, 15 cm, and 18 cm. We need to find the value of $x$ as a fraction in simplest form. 2. **Recall the property of similar polygons:** Corresponding sides of similar polygons are proportional. This means the ratio of any side in pentagon A to its corresponding side in pentagon B is the same for all pairs of corresponding sides. 3. **Identify corresponding sides:** From the problem, the side of length 15 cm in pentagon A corresponds to the side of length 15 cm in pentagon B (bottom-left side). The side of length 6 cm in pentagon A corresponds to the side of length 18 cm in pentagon B (bottom-right side). The side of length 10 cm in pentagon A corresponds to the side of length $x$ cm in pentagon B (left side). 4. **Set up ratios using known sides:** $$\frac{15}{15} = \frac{6}{18}$$ Check if these ratios are equal: $$\frac{15}{15} = 1$$ $$\frac{6}{18} = \frac{1}{3}$$ They are not equal, so we must check the correct corresponding sides or use the ratio from the two known sides in pentagon B and their corresponding sides in pentagon A. 5. **Use the two known sides in pentagon B and their corresponding sides in pentagon A to find the scale factor:** Corresponding sides: - 15 cm in B corresponds to 12 cm in A (top side) - 18 cm in B corresponds to 15 cm in A (right side) Calculate scale factors: $$k_1 = \frac{15}{12} = \frac{5}{4}$$ $$k_2 = \frac{18}{15} = \frac{6}{5}$$ Since these are not equal, check if the problem's side correspondences are correct. The problem states the sides of pentagon A as 10, 12, 15, 6, 9 and pentagon B as $x$, 15, 18. The 15 and 18 in B correspond to 9 and 6 in A respectively (bottom-left and bottom-right sides). 6. **Recalculate scale factors with correct correspondences:** - 15 cm in B corresponds to 9 cm in A - 18 cm in B corresponds to 6 cm in A Calculate scale factors: $$k_1 = \frac{15}{9} = \frac{5}{3}$$ $$k_2 = \frac{18}{6} = 3$$ These are not equal, so the sides may correspond differently. Since the pentagons are similar, the scale factor must be consistent. 7. **Try the scale factor from the two known sides in B and their corresponding sides in A:** Assuming the scale factor is $k$, then: $$k = \frac{15}{10} = \frac{3}{2}$$ $$k = \frac{15}{12} = \frac{5}{4}$$ $$k = \frac{15}{15} = 1$$ $$k = \frac{15}{6} = \frac{5}{2}$$ $$k = \frac{15}{9} = \frac{5}{3}$$ None of these match the 15 cm side in B except the 15 cm side in A (which is 15 cm). So the 15 cm side in B corresponds to the 15 cm side in A. Similarly, 18 cm side in B corresponds to 12 cm side in A: $$k = \frac{18}{12} = \frac{3}{2}$$ So the scale factor is $\frac{3}{2}$. 8. **Find $x$ using the scale factor:** The side in A corresponding to $x$ is 10 cm. Set up the proportion: $$\frac{x}{10} = \frac{3}{2}$$ 9. **Solve for $x$:** $$x = 10 \times \frac{3}{2} = 15$$ 10. **Simplify the answer:** $x = 15$ (already a whole number, simplest form). **Final answer:** $$x = 15$$