1. **State the problem:** We need to find the length of the third side (hypotenuse) of right triangles given the lengths of the two legs. 2. **Formula used:** For a right triangle with legs $a$ and $b$, and hypotenuse $c$, the Pythagorean theorem states: $$c = \sqrt{a^2 + b^2}$$ 3. **Calculate the hypotenuse for the first triangle:** Given $a = 7.5$ cm and $b = 9.7$ cm, $$c = \sqrt{7.5^2 + 9.7^2} = \sqrt{56.25 + 94.09} = \sqrt{150.34}$$ $$c \approx 12.26 \text{ cm}$$ Rounded to 1 decimal place: $12.3$ cm. 4. **Calculate the hypotenuse for the second triangle:** Given $a = 8.5$ cm and $b = 8.1$ cm, $$c = \sqrt{8.5^2 + 8.1^2} = \sqrt{72.25 + 65.61} = \sqrt{137.86}$$ $$c \approx 11.74 \text{ cm}$$ Rounded to 1 decimal place: $11.7$ cm. 5. **Calculate the missing side for the third triangle:** Given one leg $a = 6.3$ cm and the hypotenuse $c = 9.7$ cm (assuming from context), find the other leg $b$: $$b = \sqrt{c^2 - a^2} = \sqrt{9.7^2 - 6.3^2} = \sqrt{94.09 - 39.69} = \sqrt{54.4}$$ $$b \approx 7.38 \text{ cm}$$ Rounded to 1 decimal place: $7.4$ cm. **Final answers:** - First triangle hypotenuse: $12.3$ cm - Second triangle hypotenuse: $11.7$ cm - Third triangle missing side: $7.4$ cm