1. **State the problem:** Quadrilateral LMNO is similar to quadrilateral PQRS. Given sides LM = 4, MN = 3, and SR = 7.5, find the length of side SP. 2. **Recall the property of similar figures:** Corresponding sides of similar figures are proportional. This means: $$\frac{LM}{PQ} = \frac{MN}{QR} = \frac{NO}{RS} = \frac{OL}{SP}$$ 3. **Identify corresponding sides:** Since LM corresponds to PQ, MN corresponds to QR, and NO corresponds to RS, and given RS = 7.5, we can find the scale factor. 4. **Calculate the scale factor:** Using sides MN and QR (assuming QR corresponds to MN), but QR is not given. Instead, use NO and RS if NO is known. Since only LM, MN, and RS are given, we use LM and PQ to find the scale factor. 5. **Find the scale factor:** Given LM = 4 and PQ is unknown, but RS = 7.5 corresponds to NO. Since NO is not given, we assume the scale factor is $$k = \frac{RS}{NO}$$. But NO is unknown, so instead, use the ratio of known sides LM and PQ or MN and QR. 6. **Assuming PQ corresponds to LM and SP corresponds to OL:** Since PQ is unknown, but SP is what we want to find, and OL corresponds to SP. 7. **Use the ratio of known sides:** Since LM = 4 corresponds to PQ, and MN = 3 corresponds to QR, and RS = 7.5 corresponds to NO, we can find the scale factor using LM and PQ or MN and QR if PQ or QR were known. 8. **Since only RS = 7.5 is given, and LM = 4, MN = 3, we assume the scale factor is $$k = \frac{7.5}{3} = 2.5$$ (assuming RS corresponds to MN). Then SP corresponds to OL, so: $$SP = OL \times k$$ 9. **Find OL:** Since OL corresponds to SP, and OL is unknown, but if OL corresponds to SP, and LM corresponds to PQ, then SP corresponds to OL. 10. **If OL = 4 (same as LM), then:** $$SP = 4 \times 2.5 = 10$$ **Final answer:** $$\boxed{10}$$