Final answer:
To find the length of BC, we set up equations using the fact that B is the midpoint of AC, and thus AB equals BC. Solving the equations with the given values for AB and AC, we find that BC is 16 units long.
Step-by-step explanation:
The question involves finding the length of the line segment BC given that points A, B, and C are collinear and point B is the midpoint of segment AC. To find the length of BC, we can use the fact that AB and BC are equal because B is the midpoint.
Since we know that AB = 7x - 5 and AC = 5x + 17, we can express BC in terms of x as well:
- AC = AB + BC
- 5x + 17 = (7x - 5) + BC
- BC = (5x + 17) - (7x - 5)
- BC = 5x + 17 - 7x + 5
- BC = -2x + 22
Now that we have expressed BC in terms of x, we need to solve for x using the information that AC is twice the length of AB since B is the midpoint:
- AC = 2 * AB
- 5x + 17 = 2 * (7x - 5)
- 5x + 17 = 14x - 10
- 9x = 27
- x = 3
Having solved for x, we can now find the length of BC:
- BC = -2x + 22
- BC = -2(3) + 22
- BC = -6 + 22
- BC = 16
Therefore, the length of segment BC is 16 units.