The image contains a geometry problem. There is a right-angled triangle ABC, with the right angle at C. Point M is on the line extending from BC. The angle MBC is 120 degrees. The sum of the lengths of BC and AB is 36. The question asks for the lengths of AB and BC.

Solution:

• The angle ABC is supplementary to angle MBC, so angle ABC = 180° - 120° = 60°.
• In the right-angled triangle ABC, we have angle ACB = 90°, angle ABC = 60°, and angle BAC = 30°.
• We are given that BC + AB = 36.
• Using trigonometric ratios in triangle ABC:
• sin(60°) = AC / AB => AC = AB * sin(60°) = AB * (√3 / 2)
• cos(60°) = BC / AB => BC = AB * cos(60°) = AB * (1 / 2)
• Substitute BC in the given equation:
• (1/2) * AB + AB = 36
• (3/2) * AB = 36
• AB = 36 * (2/3) = 24
• Now, find BC:
• BC = (1/2) * AB = (1/2) * 24 = 12
• Check: BC + AB = 12 + 24 = 36. This matches the given information.

Final Answer:

AB = 24, BC = 12