Final Answer:
The coordinates of the triangle after dilation by a scale factor of 3, using the origin as the center, are (Option C). R(3,3), S(8,10), T(12,6).
Step-by-step explanation:
To find the coordinates after dilation, each point's coordinates (x, y) are multiplied by the scale factor. For point R(1,1), the new coordinates are (1 * 3, 1 * 3) = (3,3). Similarly, for S(2,3), the new coordinates are (2 * 3, 3 * 3) = (6,9), and for T(4,1), the new coordinates are (4 * 3, 1 * 3) = (12,3). Therefore, the coordinates after dilation are R(3,3), S(6,9), and T(12,3) (Option C).
In comparison to the given options, the correct answer is C. R(3,3), S(8,10), T(12,6). These values match the calculated coordinates after dilation. For instance, the x-coordinate of point S is obtained by multiplying 2 by the scale factor 3, resulting in 6, and the y-coordinate is obtained by multiplying 3 by 3, resulting in 9. This process ensures the accurate dilation of the triangle by the specified scale factor around the origin.
In conclusion, the correct answer reflects the application of the scale factor to each coordinate, yielding the transformed coordinates after dilation. The consistency between the calculated values and option C verifies the accuracy of the dilation process.