Final answer:
The specific angle of rotation for triangle ABC around the origin cannot be determined without additional information. However, in circular motion, tangential velocity and centripetal force vectors form a 90° angle, and a projectile's range is zero at 90° or 0°, while 45° is the optimal launch angle for maximum range.
Step-by-step explanation:
The angle of rotation in a counter-clockwise direction around the origin for triangle ABC cannot be determined without specific information about the positions of the triangle before and after the rotation. To solve such problems, one would usually compare the coordinates of the triangle's vertices before and after the rotation. In terms of general rotation angles around the origin, the standard positions are 360°, 270°, 180°, and 90°.
The angle formed between the vectors of tangential velocity and centripetal force is always 90° due to their perpendicular nature in circular motion. Similarly, the range of a projectile is zero when it's thrown vertically upwards or falls straight down, which coincides with angles of 90° or 0°. Lastly, the optimal angle of projectile motion for maximum range on level ground is 45° in the absence of air resistance.