1. **State the problem:** We have an isosceles right triangle with angles 45°, 45°, and 90°. We want to explore the relationship between the lengths of the legs and the hypotenuse. 2. **Recall the theorem:** In a 45°-45°-90° triangle, the legs are congruent, and the hypotenuse is $\sqrt{2}$ times the length of each leg. 3. **Let the length of each leg be $x$.** Then the hypotenuse length is $x\sqrt{2}$. 4. **Given:** - $AE$ and $AC$ are the legs, so $AE = AC = x$ units. - $BC$ is the hypotenuse, so $BC = x\sqrt{2}$ units. 5. **Summary:** - $AE = x$ - $AC = x$ - $BC = x\sqrt{2}$ This shows the relationship between the side lengths in an isosceles right triangle: the hypotenuse is $\sqrt{2}$ times longer than each leg.