1. **State the problem:** We need to use the special right triangle relationships for 30°-60°-90° and 45°-45°-90° triangles to find missing side lengths in given triangles. 2. **Formulas for special right triangles:** - For a 30°-60°-90° triangle, if the side opposite 30° is $x$, then the side opposite 60° is $x\sqrt{3}$ and the hypotenuse is $2x$. - For a 45°-45°-90° triangle, if each leg is $x$, then the hypotenuse is $x\sqrt{2}$. 3. **Triangle 1 (30°-60°-90°):** Side opposite 30° is 5, so: $$a = 5\sqrt{3}$$ 4. **Triangle 2 (45°-45°-90°):** Hypotenuse is 8, so each leg $a = b$ is: $$a = b = \frac{8}{\sqrt{2}}$$ Intermediate step with cancellation: $$a = b = \frac{\cancel{8}}{\cancel{\sqrt{2}}} \times \frac{\sqrt{2}}{\sqrt{2}} = 4\sqrt{2}$$ 5. **Triangle 3 (30°-60°-90°):** Hypotenuse is 12, so side opposite 30° is: $$a = \frac{12}{2} = 6$$ Side opposite 60° is: $$b = 6\sqrt{3}$$ 6. **Triangle 4 (45°-45°-90°):** Leg $a = 3\sqrt{2}$, so hypotenuse is: $$h = a\sqrt{2} = 3\sqrt{2} \times \sqrt{2} = 3 \times 2 = 6$$ Leg $b = a = 3\sqrt{2}$ 7. **Triangle 5 (Isosceles right triangle):** Altitude $a = 2\sqrt{2}$, legs $b$ equal, altitude relates to leg by: $$a = b \times \frac{\sqrt{2}}{2}$$ Solve for $b$: $$b = \frac{a}{\frac{\sqrt{2}}{2}} = 2\sqrt{2} \times \frac{2}{\sqrt{2}} = 4$$ 8. **Triangle 6 (30°-60°-90°):** Hypotenuse 14, side opposite 60° is $7\sqrt{3}$, side opposite 30° is: $$a = \frac{14}{2} = 7$$ 9. **Triangle 7 (Equilateral triangle):** All sides equal, given $b = 6\sqrt{3}$, so $a = b = 6\sqrt{3}$ 10. **Triangle 8 (30°-60°-90°):** Side adjacent 60° is 6, which is side opposite 30°, so: $$a = 6$$ Hypotenuse $b = 2a = 12$ Side opposite 60° is: $$b = 6\sqrt{3}$$ 11. **Triangle 9 (45°-45°-90°):** Side opposite 45° is 16, so legs $a = b = 16$ Hypotenuse: $$h = 16\sqrt{2}$$ **Final answers for missing sides:** - Triangle 1: $a = 5\sqrt{3}$ - Triangle 2: $a = b = 4\sqrt{2}$ - Triangle 3: $a = 6$, $b = 6\sqrt{3}$ - Triangle 4: $b = 3\sqrt{2}$ - Triangle 5: $b = 4$ - Triangle 6: $a = 7$ - Triangle 7: $a = 6\sqrt{3}$ - Triangle 8: $a = 6$, $b = 6\sqrt{3}$ - Triangle 9: $a = b = 16$ These side lengths can be matched to the letter-number grid to decode the riddle.