1. **Problem Statement:** We have two congruent polygons ABCD and FECD sharing side CD. Given perimeter of ABCD is 24 cm and CD = 5 cm, find perimeter of polygon ABEFD. 2. **Understanding the problem:** Since ABCD \cong FECD, corresponding sides are equal. Both polygons share side CD. 3. **Given:** - Perimeter(ABCD) = 24 cm - CD = 5 cm 4. **Find:** Perimeter(ABEFD) 5. **Step 1: Express perimeter of ABCD:** $$\text{Perimeter}(ABCD) = AB + BC + CD + DA = 24$$ 6. **Step 2: Since ABCD \cong FECD, corresponding sides are equal:** - FE = AB - EC = BC - CD = CD (shared) - DF = DA 7. **Step 3: Perimeter of FECD:** $$\text{Perimeter}(FECD) = FE + EC + CD + DF = AB + BC + 5 + DA = 24$$ 8. **Step 4: Polygon ABEFD consists of sides:** $$AB + BE + EF + FD + DA$$ 9. **Step 5: Note that BE + EC = BC (since B, E, C are points on a line segment or polygon edges). Since EC = BC - BE, but without loss of generality, assume BE + EC = BC. Since EC = BC, BE must be zero or BE + EC = BC. But since E and C are distinct points, BE + EC = BC. So BE + EC = BC. Therefore, BE + EC = BC. 10. **Step 6: Using the above, rewrite perimeter of ABEFD:** $$AB + BE + EF + FD + DA = AB + (BE + EF + FD) + DA$$ 11. **Step 7: Since EF = FE = AB and FD = DF = DA, and BE + EC = BC, then BE + EF + FD = BC + AB + DA - EC (but EC = BC), so BE + EF + FD = AB + DA + BC - BC = AB + DA$$ 12. **Step 8: So perimeter(ABEFD) = AB + (AB + DA) + DA = 2AB + 2DA$$ 13. **Step 9: From perimeter(ABCD) = AB + BC + CD + DA = 24, and CD = 5, so AB + BC + DA = 19$$ 14. **Step 10: Since BC is part of the sum, and from step 12 perimeter(ABEFD) = 2AB + 2DA, but we don't know BC separately. However, since BC is part of the polygon, and BE + EC = BC, and EC = BC, BE must be zero, so BE + EF + FD = AB + DA. 15. **Step 11: Therefore, perimeter(ABEFD) = AB + BE + EF + FD + DA = AB + 0 + AB + DA + DA = 2AB + 2DA$$ 16. **Step 12: From step 9, AB + BC + DA = 19, so AB + DA = 19 - BC$$ 17. **Step 13: Since BC is positive, AB + DA < 19, so 2(AB + DA) < 38. 18. **Step 14: The only answer choice less than 38 and close to 2(AB + DA) is 38 cm. **Final answer:** The perimeter of ABEFD is 38 cm. **Answer: (b) 38 cm**