1. **State the problem:** We have a right triangle with angles 30º, 60º, and 90º. The side opposite the 30º angle is given as $9\sqrt{3}$ mm, and we need to find the length $c$ opposite the 60º angle. 2. **Recall the properties of a 30º-60º-90º triangle:** The sides are in the ratio $1 : \sqrt{3} : 2$, where: - The side opposite 30º is the shortest side (let's call it $x$). - The side opposite 60º is $x\sqrt{3}$. - The hypotenuse is $2x$. 3. **Identify the given side:** The side opposite 30º is $9\sqrt{3}$ mm, so: $$x = 9\sqrt{3}$$ 4. **Find the side opposite 60º:** Using the ratio, the side opposite 60º is: $$c = x\sqrt{3} = 9\sqrt{3} \times \sqrt{3}$$ 5. **Simplify the expression:** $$c = 9 \times \cancel{\sqrt{3} \times \sqrt{3}} = 9 \times 3 = 27$$ 6. **Final answer:** $$c = 27 \text{ mm}$$ So, the length $c$ opposite the 60º angle is 27 millimeters.