**Core Concept**
The treatment of hyperthyroidism often involves the use of radioactive isotopes to destroy thyroid tissue. The choice of isotope depends on the specific indication, with some isotopes being more suitable for diagnostic purposes while others are used therapeutically.

**Why the Correct Answer is Right**
Radioiodine therapy is a common treatment for hyperthyroidism, particularly for conditions such as Graves' disease. The radioisotope of iodine used for thyroid tissue destruction is **I-131**, which emits beta particles that damage the thyroid tissue. This leads to a reduction in thyroid hormone production and can help to alleviate symptoms of hyperthyroidism. The use of I-131 is based on the fact that the thyroid gland selectively takes up iodine from the bloodstream, allowing the radioactive isotope to concentrate in the thyroid tissue and cause damage.

**Why Each Wrong Option is Incorrect**
* **Option A:** I-123 is a radioactive isotope of iodine, but it is primarily used for diagnostic purposes due to its shorter half-life and lower energy emissions, making it less suitable for therapeutic use.
* **Option B:** I-125 is another radioactive isotope of iodine, but it has a longer half-life and is often used for brachytherapy and other applications, rather than thyroid tissue destruction.
* **Option D:** I-131 is the correct answer, but I-129 is a stable isotope of iodine and is not radioactive, making it an incorrect choice.

**Clinical Pearl / High-Yield Fact**
When using radioiodine therapy, it is essential to note that the effectiveness of treatment can be influenced by factors such as the dose of I-131 administered, the patient's thyroid function, and the presence of any underlying thyroid nodules or cancer.

**Correct Answer: C. I-131**