1. **Problem Statement:** We have two similar pentagons QRSTU and HGFEI. Given some side lengths, we need to find the lengths of RS and HI. 2. **Given Data:** - Pentagon QRSTU: QR = 40, QU = 48, UT = 56, RS and ST unknown. - Pentagon HGFEI: GH = 15, GF = 9, FE, EI, and HI unknown. 3. **Key Concept:** Since the pentagons are similar, corresponding sides are proportional. That means the ratio of any side in QRSTU to its corresponding side in HGFEI is constant. 4. **Identify Corresponding Sides:** - QR corresponds to GH - QU corresponds to GI (not given, but we can use QR and GH for ratio) 5. **Calculate the scale factor:** $$\text{scale factor} = \frac{QR}{GH} = \frac{40}{15} = \frac{8}{3} \approx 2.6667$$ 6. **Find RS:** - RS corresponds to GF (given as 9) - Using the scale factor: $$RS = GF \times \text{scale factor} = 9 \times \frac{8}{3} = 24$$ 7. **Find HI:** - HI corresponds to ST (unknown) - We know UT = 56 corresponds to EI (unknown), but we don't have EI or ST. - However, since QU = 48 corresponds to GI (not given), we cannot directly find HI without more info. 8. **Assuming the order of vertices is consistent, HI corresponds to ST. Since ST is not given, but UT = 56 corresponds to EI, and we have no EI, we cannot find HI directly.** 9. **Alternative approach:** If we assume the pentagons are labeled in order, then HI corresponds to ST. Since ST is not given, but we can find ST using the perimeter or other info if available. Since no other info is given, we cannot find HI. **Final answers:** $$RS = 24$$ $$HI = \text{Cannot be determined with given data}$$