Given the points a (0, 2) and b (2, 3), and the points a' (0, 4) and b' (4, 6), we need to compare the lengths of the initial segment ab and the new segment a'b' to determine the dilation scale factor.
First, let's calculate the distance between points a and b which forms the segment ab. This is done using the distance formula: √[(x2 - x1)² + (y2 - y1)²]. By substituting the given coordinates into the formula, we have √[(2 - 0)² + (3 - 2)²], which results in approximately 2.236.
Next, we find the length of the new segment a'b' in the same manner. Using the coordinates of a' and b', we find this to be √[(4 - 0)² + (6 - 4)²], which is approximately 4.472.
To find by what scale factor segment ab was dilated to form segment a'b', we divide the length of a'b' by the length of ab, i.e. 4.472/2.236. After performing the division, we find the scale factor to be 2.0.
Based on the given options (one half, 2, 3, 4), it appears that the correct scale factor is 2. Hence, segment ab was dilated by a factor of 2 to create segment a'b'.