1. **Problem Statement:** Find the hypotenuse of the top right-angled triangle with legs 5 cm and 5 cm. Also, find all missing lengths of the stacked right-angled triangles from bottom to top. 2. **Formula Used:** For a right-angled triangle with legs $a$ and $b$, the hypotenuse $c$ is given by the Pythagorean theorem: $$c = \sqrt{a^2 + b^2}$$ 3. **Step-by-step solution:** - **Top triangle (legs 5 cm and 5 cm):** $$c = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \approx 7.07\text{ cm}$$ - **Next triangle (legs 14 cm and 11 cm):** $$c = \sqrt{14^2 + 11^2} = \sqrt{196 + 121} = \sqrt{317} \approx 17.8\text{ cm}$$ - **Large magenta triangle (one leg 16 cm, hypotenuse unknown):** Assuming the other leg is given or can be found, but since only one leg and hypotenuse unknown, if the other leg is missing, we cannot find hypotenuse without more info. - **Large blue triangle (legs 12 cm and 5 cm):** $$c = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\text{ cm}$$ - **Orange and purple triangles near bottom:** - Triangle with legs 7 cm and 4 cm: $$c = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65} \approx 8.06\text{ cm}$$ - Triangle with legs 6 cm and 9 cm: $$c = \sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117} \approx 10.82\text{ cm}$$ - Triangle with legs 2 cm and 6 cm: $$c = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \approx 6.32\text{ cm}$$ 4. **Summary of hypotenuses:** - Top triangle: $5\sqrt{2} \approx 7.07$ cm - Triangle with legs 14 cm and 11 cm: $\approx 17.8$ cm - Large blue triangle (12 cm, 5 cm): 13 cm - Triangle (7 cm, 4 cm): $\approx 8.06$ cm - Triangle (6 cm, 9 cm): $\approx 10.82$ cm - Triangle (2 cm, 6 cm): $2\sqrt{10} \approx 6.32$ cm 5. **Note:** For the large magenta triangle with one leg 16 cm and unknown hypotenuse, more information is needed to find the missing length. This completes the calculation of all missing hypotenuses from bottom to top where possible.