1. **Stating the problem:** We are given two triangles with right angles and some side lengths. We need to find the missing sides or verify the given sides using the Pythagorean theorem. 2. **Recall the Pythagorean theorem:** For a right triangle with legs $a$ and $b$, and hypotenuse $c$, the relation is: $$c^2 = a^2 + b^2$$ 3. **Problem b):** Given $b = 4.5$ cm, $c = 4.5$ cm, and angle $\alpha = 90^\circ$ between them. Since $\alpha$ is the right angle between sides $b$ and $c$, these are the legs of the triangle, and the hypotenuse $a$ is unknown. Using the Pythagorean theorem: $$a^2 = b^2 + c^2$$ Substitute values: $$a^2 = (4.5)^2 + (4.5)^2 = 20.25 + 20.25 = 40.5$$ Calculate $a$: $$a = \sqrt{40.5} \approx 6.36 \text{ cm}$$ 4. **Problem d):** Given $a = 8.2$ cm, $b = 5.2$ cm, and angle $\gamma = 90^\circ$. Here, $\gamma$ is the right angle between sides $a$ and $b$, so these are legs, and hypotenuse $c$ is unknown. Using the Pythagorean theorem: $$c^2 = a^2 + b^2$$ Substitute values: $$c^2 = (8.2)^2 + (5.2)^2 = 67.24 + 27.04 = 94.28$$ Calculate $c$: $$c = \sqrt{94.28} \approx 9.71 \text{ cm}$$ **Final answers:** - For problem b): $a \approx 6.36$ cm - For problem d): $c \approx 9.71$ cm