1. **Problem:** Find the length of the hypotenuse $c$ in a right triangle given the legs. 2. **Formula:** Use the Pythagorean theorem: $$c = \sqrt{a^2 + b^2}$$ where $a$ and $b$ are the legs of the triangle. 3. **Step-by-step solution for part (a):** - Given legs: $1$ and $2$ - Calculate: $$c = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$$ - So, the exact length of the hypotenuse is $\sqrt{5}$. This method applies similarly to all parts in question 3. --- **Question 3 (a):** $$c = \sqrt{1^2 + 2^2} = \sqrt{5}$$ **Question 3 (b):** $$c = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}$$ **Question 3 (c):** $$c = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}$$ **Question 3 (d):** $$c = \sqrt{6^2 + 1^2} = \sqrt{36 + 1} = \sqrt{37}$$ **Question 3 (e):** $$c = \sqrt{10^2 + 3^2} = \sqrt{100 + 9} = \sqrt{109}$$ **Question 3 (f):** $$c = \sqrt{8^2 + 17^2} = \sqrt{64 + 289} = \sqrt{353}$$ --- **Summary:** - The hypotenuse lengths for question 3 are: - (a) $\sqrt{5}$ - (b) $\sqrt{58}$ - (c) $\sqrt{34}$ - (d) $\sqrt{37}$ - (e) $\sqrt{109}$ - (f) $\sqrt{353}$ These are exact values as requested. --- **Note:** For the other questions (4, 5, 6), since the user asked to solve only the first question, they are not solved here.