1. **Problem statement:** Given a right isosceles triangle ABE with hypotenuse AE = 10 cm, find the area of the square whose side is the leg BE of the triangle. 2. **Formula and rules:** In a right isosceles triangle, the legs are equal, and the hypotenuse $AE$ relates to the leg $BE$ by the Pythagorean theorem: $$AE = BE \sqrt{2}$$ 3. **Calculate the leg BE:** $$BE = \frac{AE}{\sqrt{2}} = \frac{10}{\sqrt{2}}$$ 4. **Rationalize the denominator:** $$BE = \frac{10}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$$ 5. **Area of the square with side BE:** $$\text{Area} = BE^2 = (5\sqrt{2})^2 = 25 \times 2 = 50$$ 6. **Final answer:** The area of the square is $50$ square centimeters.