1. **Problem statement:** Calculate the side length of the equilateral triangle inscribed in an isosceles right triangle with base $c=12.0$ cm as shown in Abb. 3.150 a). 2. **Understanding the problem:** We have an isosceles right triangle with legs $a$, $a$ and base $c=12.0$ cm. An equilateral triangle is inscribed inside it in a specific position. 3. **Known facts and formulas:** - In an isosceles right triangle, the legs are equal: $a = a$. - The base $c$ is the hypotenuse of the right triangle, so by Pythagoras: $$c = a\sqrt{2}$$ - The equilateral triangle has all sides equal, denote its side length as $s$. 4. **Calculate leg length $a$ of the right triangle:** $$a = \frac{c}{\sqrt{2}} = \frac{12.0}{\sqrt{2}} = 12.0 \times \frac{\sqrt{2}}{2} = 6\sqrt{2}$$ 5. **Relate the equilateral triangle side $s$ to $a$:** From the figure and problem context (Abb. 3.150 a)), the equilateral triangle is inscribed such that its side $s$ fits along the legs $a$ of the right triangle. 6. **Using geometric relations:** The equilateral triangle side $s$ equals the leg length $a$: $$s = a = 6\sqrt{2}$$ 7. **Final answer:** $$\boxed{s = 6\sqrt{2} \approx 8.49\text{ cm}}$$ This is the side length of the equilateral triangle inscribed as in Abb. 3.150 a).