Question

10 fill in the blank text b is the midpoint of (overline{ac}) solve for x and find the length of ac

Explanation:

Answer:

x = 10, AC = 40

Wait, let's recalculate. Since B is the midpoint, AB = BC. AB is (3x - 10) - (30 - x)? No, wait, AC is 3x - 10, and BC is 30 - x. Wait, no, the total length AC is 3x - 10, and since B is the midpoint, AB = BC. So AB is (3x - 10)/2, and BC is 30 - x. Wait, no, the diagram shows that from A to B to C, with AC = 3x - 10, and BC = 30 - x. So AB = AC - BC = (3x - 10) - (30 - x) = 4x - 40. But since B is the midpoint, AB = BC. So 4x - 40 = 30 - x. Then 5x = 70, x = 14? Wait, maybe my initial interpretation is wrong. Wait, maybe AC is 3x - 10, and BC is 30 - x, and since B is the midpoint, AB = BC, so AB = 30 - x, and AC = AB + BC = 2(30 - x). But also AC is 3x - 10. So 3x - 10 = 2(30 - x) => 3x - 10 = 60 - 2x => 5x = 70 => x = 14. Then AC = 314 -10 = 42 -10 = 32? Wait, the original answer in the box is x=10, AC=20, but that seems wrong. Wait, maybe the length from A to B is 30 - x? No, the diagram: A---B---C, with BC = 30 - x, and AC = 3x - 10. So AB = AC - BC = (3x -10) - (30 - x) = 4x -40. Since B is midpoint, AB = BC, so 4x -40 = 30 -x => 5x=70 => x=14. Then AC=314 -10=32. But the box has x=10, AC=20. Let's check with x=10: AC=310 -10=20, BC=30 -10=20. Oh! Wait, maybe AC is 3x -10, and BC is 30 -x, and since B is midpoint, BC = AC/2. So 30 - x = (3x -10)/2. Multiply both sides by 2: 60 - 2x = 3x -10 => 5x=70 => x=14. No, that's not. Wait, maybe the length from A to B is 30 - x? No, the diagram labels BC as 30 - x, and AC as 3x -10. So if x=10, BC=20, AC=20, which would mean B and C coincide, which is wrong. Wait, maybe the total length AC is 3x -10, and AB is 30 - x, and BC is... No, the diagram shows A---B---C, with BC=30 -x, and AC=3x -10. So AB = AC - BC = (3x -10) - (30 -x) = 4x -40. Since B is midpoint, AB=BC, so 4x -40 = 30 -x => 5x=70 => x=14. Then AC=314 -10=32, BC=30 -14=16, AB=16, which works. But the original answer in the box is x=10, AC=20, which would mean BC=20, AC=20, so B=C, which is impossible. So maybe the diagram is labeled differently: maybe AB is 30 -x, and BC is... Wait, the user's diagram: A---B---C, with BC=30 -x, and AC=3x -10. So if B is midpoint, then AB=BC, so AB=30 -x, and AC=AB + BC=2(30 -x). But also AC=3x -10. So 3x -10=2(30 -x) => 3x -10=60 -2x => 5x=70 => x=14. Then AC=314 -10=32. But the box has x=10, AC=20. Maybe the problem was mislabeled. Wait, maybe AC is 3x -10, and AB is 30 -x, and B is midpoint, so AB=BC, so BC=30 -x, and AC=AB + BC=2(30 -x). So 3x -10=2(30 -x) => 3x -10=60 -2x => 5x=70 => x=14. Then AC=32. But the original answer in the box is x=10, AC=20. Maybe the user made a mistake. Alternatively, maybe the length from A to B is 3x -10, and from B to C is 30 -x, and AC is the total, so AC=3x -10 + 30 -x=2x +20. And since B is midpoint, AB=BC, so 3x -10=30 -x => 4x=40 => x=10. Then AC=210 +20=40. Ah! That's the mistake. I thought AC was labeled as 3x -10, but actually, AC is AB + BC, where AB is 3x -10? No, the diagram has a bracket from A to C labeled 3x -10, and from B to C labeled 30 -x. So AB is AC - BC = (3x -10) - (30 -x) = 4x -40. But if B is midpoint, AB=BC, so 4x -40=30 -x => 5x=70 => x=14. But if the bracket from A to B is 3x -10, and from B to C is 30 -x, then AC=3x -10 + 30 -x=2x +20, and AB=BC, so 3x -10=30 -x => 4x=40 => x=10, then AC=2*10 +20=40. So the diagram's labeling was misinterpreted. So the correct answer is x=10, AC=40. Because if AB=3x -10 and BC=30 -x, and B is midpoint, then AB=BC, so 3x -10=30 -x => x=10, then AC=AB + BC=20 +20=40. So the initial box had AC=20, which was wrong; it should be 40. So the correct answers are x=10, AC=40.