1. **Problem statement:** Find the missing side length, area, and perimeter of a right isosceles triangle with one leg length $\sqrt{2}$ meters and one angle of 45°. 2. **Identify the triangle type and properties:** Since the triangle has a right angle and a 45° angle, it is a 45°-45°-90° triangle, which is an isosceles right triangle. 3. **Use the 45°-45°-90° triangle side ratio rule:** The sides are in the ratio $1:1:\sqrt{2}$, where the legs are equal and the hypotenuse is $\sqrt{2}$ times a leg. 4. **Find the missing side lengths:** Given one leg $= \sqrt{2}$ meters, the other leg is also $\sqrt{2}$ meters. The hypotenuse $h$ is: $$ h = \sqrt{2} \times \sqrt{2} = \sqrt{2 \times 2} = \sqrt{4} = 2 $$ 5. **Calculate the perimeter:** $$ \text{Perimeter} = \text{leg}_1 + \text{leg}_2 + \text{hypotenuse} = \sqrt{2} + \sqrt{2} + 2 = 2\sqrt{2} + 2 $$ 6. **Calculate the area:** Area of a right triangle is: $$ \text{Area} = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2 = \frac{1}{2} \times \sqrt{2} \times \sqrt{2} = \frac{1}{2} \times 2 = 1 $$ **Final answers:** - Missing side lengths: other leg $= \sqrt{2}$ m, hypotenuse $= 2$ m - Perimeter $= 2\sqrt{2} + 2$ m - Area $= 1$ m²