1. **Problem Statement:** We have a right-angled triangle ABC with D and E as midpoints of AB and BC respectively. We want to determine which statements are sufficient to find the length of AC. 2. **Given Statements:** - I. AE = 19 - II. CD = 22 - III. Angle B is a right angle. 3. **Key Points:** - Since D and E are midpoints, AD = DB and BE = EC. - Angle B is right angle means triangle ABC is right-angled at B. 4. **Analyzing Statements:** - Statement III alone tells us angle B is 90°, so by Pythagoras theorem: $$AC^2 = AB^2 + BC^2$$ - Statement I gives AE = 19. Since E is midpoint of BC, AE is a segment connecting A to midpoint E. - Statement II gives CD = 22. Since D is midpoint of AB, CD is a segment connecting C to midpoint D. 5. **Using Midpoint Theorem:** - Segment DE is parallel to AC and DE = 1/2 AC. - Lengths AE and CD relate to the triangle but do not directly give AC without additional info. 6. **Checking combinations:** - I and II alone: Knowing AE and CD does not give enough info to find AC. - III alone: Knowing angle B is right angle allows use of Pythagoras but we need side lengths. - I and III: Knowing AE and right angle B still lacks direct side lengths. - II and III: Knowing CD and right angle B can help because CD relates to sides AB and BC. - All three statements: Combining AE, CD, and right angle B gives enough info to find AC. **Final conclusion:** Only "All three statements" together are sufficient to determine the length of AC. **Answer:** All three statements.