Manhatten Prep Geometry: Why does this approach not work?

This problem is from MANHATTAN PREP Guide 3 Geometry Hard Practice Question Set Question 2. I followed the following approach to solve this problem:
• Find the length of AB = 5
• Find the length of BC = 2AB = 2x5 = 10
• Connect the points B and C, and C and A to form a triangle ABC.
• Based on the lengths of 2 sides (AB and BC), estimate the length of AC to be in the range 5 < x < 15

• Then I calculated the distance from point A (0, 0) to each of the options:

  • Option A: 5.3
  • Option B: 15
  • Option C: 14.86
  • Option D: 14.86
  • Option E: 13.6

Since only the length from Option B is out of range for the length of AC, I chose option B as the answer. But the guide says that the answer is C. It uses the value of length BC and eliminates all the options that result in the distance of 10. I understand the approach provided in the guide but shouldn’t the approach I used also work? What did I do wrong?

Question:
In the coordinate plane above, point C is not displayed. If the length of line segment BC is twice the length of line segment AB, which of the following could not be the coordinates of point C?

Formula for distance between 2 points is Root((x2-x1)^2 + (y2-y1)^2), You got length of AB correct as 5 so now length of BC should be 10. This means than 100 = (difference(x2,x1)^2 + difference(y2,y1)^2) based on formula

You have one set of x,y as 3,4 and you can use other sets from options to eliminate others based above highlighted equation

For C only, above equation doesn’t holds true as ((10-3)^2+(11-4)^2) = (49+49) which is not equal to 100. Not sure about technique but this is how I figured it out

The technique you described is already given in the book and I fully understand it. I am just trying to figure out if any of the assumptions I made when solving the problem by my method are wrong (so that I don’t make the same mistake when solving other problems by using the same assumptions).