1. **Problem:** Calculate the exact area of an equilateral triangle with each side length 24 cm, which is used as a triangular bracket glued to the back of a circular mirror of radius 14 cm. 2. **Formula:** The area $A$ of an equilateral triangle with side length $a$ is given by: $$A = \frac{\sqrt{3}}{4} a^2$$ 3. **Calculation:** Substitute $a = 24$ cm: $$A = \frac{\sqrt{3}}{4} \times 24^2$$ 4. Simplify the expression: $$A = \frac{\sqrt{3}}{4} \times 576$$ 5. Calculate the multiplication: $$A = 144 \sqrt{3}$$ 6. **Answer:** The exact area of the triangular bracket is: $$\boxed{144 \sqrt{3} \text{ cm}^2}$$ This area is exact and does not depend on the radius of the mirror for this problem.