1. **State the problem:** Given a right triangle with legs $a$ and $b$ and hypotenuse $c$, where $a = 2\sqrt{3}$ and $c = 2b$, find the length of side $b$. 2. **Recall the Pythagorean theorem:** For a right triangle, the relationship between the sides is: $$c^2 = a^2 + b^2$$ 3. **Substitute the given values:** We know $a = 2\sqrt{3}$ and $c = 2b$, so: $$ (2b)^2 = (2\sqrt{3})^2 + b^2 $$ 4. **Simplify each term:** $$ 4b^2 = 4 \times 3 + b^2 $$ $$ 4b^2 = 12 + b^2 $$ 5. **Isolate $b^2$:** $$ 4b^2 - b^2 = 12 $$ $$ 3b^2 = 12 $$ 6. **Divide both sides by 3:** $$ \cancel{3}b^2 = \cancel{3}4 $$ $$ b^2 = 4 $$ 7. **Take the square root of both sides:** $$ b = \sqrt{4} = 2 $$ **Final answer:** $$ b = 2 $$