Final answer:
The twins are in the fetal stage of prenatal development, likely between weeks 9 to 12, as sexual differentiation of the genitalia allows for accurate determination of their genders via ultrasound.
Step-by-step explanation:
The doctor's ability to determine that a pregnant woman is expecting a boy and a girl indicates that the twins' sexual differentiation has occurred. This takes place during the fetal stage of prenatal development, specifically between weeks 9 to 12, when the external genitals of male and female fetuses become distinct. Since the biological sex can be determined with almost 100 percent accuracy using obstetric ultrasound by week 12, the twins are likely within this period of development or beyond. By this stage, critical development such as the initial formation of internal organs and external features like fingers, toes, and fingerprints has also occurred, making the fetus appear more like a small human.