1. **Problem statement:** We have a right triangle with legs of lengths 2 units and 4 units. We want to find the area of the square that shares its side with the hypotenuse (the third side) of this triangle. 2. **Step 1: Find the length of the hypotenuse.** Since the triangle is right-angled, we use the Pythagorean theorem: $$c = \sqrt{a^2 + b^2}$$ where $a = 2$ units and $b = 4$ units. 3. **Calculate:** $$c = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20}$$ 4. **Simplify the square root:** $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ 5. **Step 2: Find the area of the square.** The square's side length is equal to the hypotenuse length $c = 2\sqrt{5}$. Area of a square is given by: $$\text{Area} = \text{side}^2$$ 6. **Calculate the area:** $$\text{Area} = (2\sqrt{5})^2 = 2^2 \times (\sqrt{5})^2 = 4 \times 5 = 20$$ **Final answer:** The area of the square is $20$ units$^2$.