1. **Problem Statement:** Find the length of the hypotenuse $c$ in each right-angled triangle using the Pythagorean theorem. 2. **Formula:** The Pythagorean theorem states: $$c = \sqrt{a^2 + b^2}$$ where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse. 3. **Important Rule:** The hypotenuse is always the longest side opposite the right angle. --- ### Exercise 3A Question 1 **a.** Given $a=24$, $b=10$, find $c$: $$c = \sqrt{24^2 + 10^2} = \sqrt{576 + 100} = \sqrt{676} = 26$$ **b.** Given $a=8$, $b=15$, find $c$: $$c = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$$ **c.** Given $a=12$, $b=9$, find $c$: $$c = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15$$ **d.** Given $a=24$, $b=7$, find $c$: $$c = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25$$ **e.** Given $a=9$, $b=40$, find $c$: $$c = \sqrt{9^2 + 40^2} = \sqrt{81 + 1600} = \sqrt{1681} = 41$$ **f.** Given $a=14$, $b=48$, find $c$: $$c = \sqrt{14^2 + 48^2} = \sqrt{196 + 2304} = \sqrt{2500} = 50$$ --- ### Exercise 3A Question 2 (correct to two decimal places) **a.** Given $a=4$, $b=2$, find $c$: $$c = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47$$ **b.** Given $a=3$, $b=1$, find $c$: $$c = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.16$$ **c.** Given $a=12$, $b=10$, find $c$: $$c = \sqrt{12^2 + 10^2} = \sqrt{144 + 100} = \sqrt{244} \approx 15.62$$ **d.** Given $a=8.6$, $b=7.4$, find $c$: $$c = \sqrt{8.6^2 + 7.4^2} = \sqrt{73.96 + 54.76} = \sqrt{128.72} \approx 11.35$$ **e.** Given one leg $a=5$ and hypotenuse $c$, find the other leg $b$ (assuming right angle): Since $c$ is hypotenuse, if $b$ is unknown, we need more info. If $c$ is unknown, and $a=5$, and the other leg is unknown, we cannot find $c$ without the other leg. **f.** Given $a=0.04$, $b=0.14$, find $c$: $$c = \sqrt{0.04^2 + 0.14^2} = \sqrt{0.0016 + 0.0196} = \sqrt{0.0212} \approx 0.15$$ --- **Summary:** Use the Pythagorean theorem to find the hypotenuse by squaring both legs, adding them, and taking the square root. For unknown legs, rearrange the formula accordingly. Always round to two decimal places when required.