Final answer:
Star A is half the distance from the observer compared to Star B because it appears 4 times brighter and both stars have the same luminosity. This is related to the inverse-square law of light.
Step-by-step explanation:
If two stars, Star A and Star B, have the same luminosity but different apparent brightnesses, the difference in brightness is due to their distances from the observer. According to the inverse-square law for light, the apparent brightness of a star is inversely proportional to the square of its distance from the observer. Thus, if Star A appears 4 times brighter than Star B, and they have the same luminosity, then Star A must be closer to the observer.
Using the inverse-square law, which can be represented by the formula brightness ≈ 1/(distance2), we can find that Star A is half the distance from the observer as compared to Star B. This is because to produce 4 times the brightness (4 being the square of 2), Star A must be at 1/2 the distance of Star B. Therefore, if Star B is at distance d, then Star A would be at distance d/2 from us.