1. **Problem statement:** Find the unknown side lengths in the right triangles using the Pythagorean theorem, giving answers in surd form. 2. **Formula:** For a right triangle with sides $a$, $b$, and hypotenuse $c$, the Pythagorean theorem states: $$c^2 = a^2 + b^2$$ If the unknown side is a leg, use: $$a = \sqrt{c^2 - b^2}$$ If the unknown side is the hypotenuse, use: $$c = \sqrt{a^2 + b^2}$$ 3. **Solve each part:** **a.** Given sides 1 m, 3 m, and unknown $x$ m (hypotenuse): $$x = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}$$ **b.** Given sides 4 cm, 10 cm, and unknown $y$ cm (leg): $$y = \sqrt{10^2 - 4^2} = \sqrt{100 - 16} = \sqrt{84} = 2\sqrt{21}$$ **c.** Given sides 48 km, 125 km, and unknown $m$ km (leg): $$m = \sqrt{125^2 - 48^2} = \sqrt{15625 - 2304} = \sqrt{13321}$$ Since $13321 = 115^2 + 6^2$ is not a perfect square, leave as $\sqrt{13321}$. **d.** Given sides 7 cm, 6 cm, and unknown $y$ cm (hypotenuse): $$y = \sqrt{7^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85}$$ **e.** Given sides 110 m, 326 m, and unknown $x$ m (hypotenuse): $$x = \sqrt{110^2 + 326^2} = \sqrt{12100 + 106276} = \sqrt{118376}$$ **f.** Given sides 4 cm, 12 cm, and unknown $x$ cm (hypotenuse): $$x = \sqrt{4^2 + 12^2} = \sqrt{16 + 144} = \sqrt{160} = 4\sqrt{10}$$ 4. **Final answers:** - a: $x = \sqrt{10}$ m - b: $y = 2\sqrt{21}$ cm - c: $m = \sqrt{13321}$ km - d: $y = \sqrt{85}$ cm - e: $x = \sqrt{118376}$ m - f: $x = 4\sqrt{10}$ cm These are the unknown sides in surd form.