1. The problem asks to find the length of the hypotenuse for right triangles given the lengths of the legs. 2. We use the Pythagorean theorem formula for right triangles: $$c = \sqrt{a^2 + b^2}$$ where $c$ is the hypotenuse and $a$, $b$ are the legs. 3. Let's solve part (a) as an example: Given legs: $a=4$ cm, $b=8$ cm. Calculate: $$c = \sqrt{4^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80}$$ Simplify $\sqrt{80}$: $$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \approx 8.9$$ So, the hypotenuse is approximately 8.9 cm. 4. For part (b), legs are 5 m and 9 m: $$c = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106} \approx 10.3$$ 5. For part (c), legs are 10 in and 6 in: $$c = \sqrt{10^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.7$$ 6. For part (d), legs are 7 yd and 11 yd: $$c = \sqrt{7^2 + 11^2} = \sqrt{49 + 121} = \sqrt{170} \approx 13.0$$ 7. For part (e), legs are 6 ft and 12 ft: $$c = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180} \approx 13.4$$ 8. For part (f), legs are 5 km and 13 km is the hypotenuse, so to find the missing leg $b$: $$b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$$ 9. For question 2, legs are 11 cm and 10 cm: $$c = \sqrt{11^2 + 10^2} = \sqrt{121 + 100} = \sqrt{221} \approx 14.9$$ Thus, the hypotenuse is approximately 14.9 cm. This method applies to all right triangles: square the legs, add, then take the square root to find the hypotenuse.