1. **Stating the problem:** We have a right triangle with angles 45°, 45°, and 90°. We want to understand the properties of this triangle. 2. **Formula and important rules:** In a 45-45-90 triangle, the sides are in the ratio $1:1:\sqrt{2}$. 3. **Explanation:** Since the two legs are equal, if each leg has length $x$, then the hypotenuse has length $x\sqrt{2}$. 4. **Intermediate work:** If one leg is $x$, the hypotenuse is calculated as: $$\text{hypotenuse} = x\sqrt{2}$$ 5. **Summary:** The 45-45-90 triangle is an isosceles right triangle where the legs are equal and the hypotenuse is $\sqrt{2}$ times the length of each leg.