Final answer:
The question requires using the distance formula to identify the value of x where the distance between points A and B is equal to the distance between B and C. Upon solving, we find that x = -1, which is not listed in the provided options, indicating a potential error in the problem or options given.
Step-by-step explanation:
The question is asking to find the value of x such that the distance between points A and B is equal to the distance between points B and C, where A = (-3,-2), B = (x, 3), and C = (4,5). We can use the distance formula which is derived from the Pythagorean theorem to find the distances AB and BC:
For AB: √[(x - (-3))^2 + (3 - (-2))^2] = distance AB,
For BC: √[(4 - x)^2 + (5 - 3)^2] = distance BC.
Because AB = BC, we have:
- (x + 3)^2 + 5^2 = (4 - x)^2 + 2^2,
- x^2 + 6x + 9 + 25 = 16 - 8x + x^2 + 4,
- 6x + 8x = 16 + 4 - 9 - 25,
- 14x = -14,
- x = -1.
However, -1 is not one of the options provided, which could mean either a typo in the problem or the options. Since the options presented are 0, 1, 2, and 3, and none of them match the calculated value of x, we should recheck the calculations or clarify the question.