**Core Concept**
Iodine radioisotopes are used in the treatment of hyperthyroidism due to their ability to selectively target thyroid tissue. The radioisotope is taken up by the thyroid gland, where it emits radiation that destroys thyroid tissue, thereby reducing thyroid hormone production.

**Why the Correct Answer is Right**
The correct answer is Iodine-131 (I-131). I-131 is a radioactive isotope of iodine that is selectively taken up by the thyroid gland. Once inside the thyroid gland, I-131 emits beta radiation that destroys thyroid tissue, leading to a reduction in thyroid hormone production. This makes I-131 a highly effective treatment for hyperthyroidism, particularly in cases of Graves' disease and thyroid cancer.

**Why Each Wrong Option is Incorrect**
* **Option A:** Iodine-123 (I-123) is a radioactive isotope of iodine used primarily in nuclear medicine for imaging the thyroid gland. It is not used for therapeutic purposes due to its short half-life and low radiation dose.
* **Option B:** Iodine-125 (I-125) is a radioactive isotope of iodine used in brachytherapy for prostate cancer and other cancers. It is not used in the treatment of hyperthyroidism.
* **Option C:** Iodine-129 (I-129) is a radioactive isotope of iodine with a long half-life, making it unsuitable for therapeutic use in humans.

**Clinical Pearl / High-Yield Fact**
Remember that I-131 should be avoided in pregnant women and breastfeeding mothers due to the risk of radiation exposure to the fetus and infant.

**Correct Answer:** C. Iodine-131 (I-131)