Objective: Build instant visual recognition of 3-4-5, 5-12-13, 30-60-90, and 45-45-90 triangles to save critical minutes on geometry questions.Questions 1. A right triangle has legs of length 6 and 8. What is the length of the hypotenuse? 2. A 30-60-90 triangle has a hypotenuse of length 12. What is the length of its shortest side? 3. An isosceles right triangle has a hypotenuse of \(5\sqrt{2}\). What is the length of one leg? 4. A right triangle has a leg of 5 and a hypotenuse of 13. What is the length of the other leg? 5. In a 30-60-90 triangle, the side opposite the 60° angle is \(9\sqrt{3}\). What is the length of the hypotenuse? 6. The perimeter of a square is 32. What is the length of its diagonal? 7. A right triangle has legs of 9 and 12. What is its perimeter? 8. In a right triangle, \(\sin(\theta) = \frac{3}{5}\). If the hypotenuse is 20, what is the length of the side adjacent to θ? 9. A 30-60-90 triangle has a shortest side of 15 cm. What is its exact perimeter? 10. The hypotenuse of a 30-60-90 triangle is 72 units. What is its exact perimeter?
🔑 Answer Key & Explanations 1. 10 → This is a scaled 3-4-5 triple multiplied by 2 (6-8-10). 2. 6 → In a 30-60-90 triangle, the shortest side is always exactly half of the hypotenuse (\(\frac{12}{2} = 6\)). 3. 5 → A 45-45-90 triangle follows the side ratio \(x : x : x\sqrt{2}\). Since the hypotenuse is \(5\sqrt{2}\), x = 5. 4. 12 → This is a standard 5-12-13 Pythagorean triple. 5. 18 → The side opposite 60° is \(x\sqrt{3}\), so the short side x = 9. The hypotenuse is 2x = 18. 6. \(8\sqrt{2}\) → A square split by a diagonal forms two 45-45-90 triangles. Side length \(= \frac{32}{4} = 8\). Diagonal \(= 8\sqrt{2}\). 7. 36 → This is a 3-4-5 triple multiplied by 3 (9-12-15). Perimeter = 9 + 12 + 15 = 36. 8. 16 → \(\sin = \frac{\text{opposite}}{\text{hypotenuse}}\). A triangle with a hypotenuse of 20 scales a 3-4-5 triangle up by 4 (12-16-20). The adjacent leg is 16. 9. \(45 + 15\sqrt{3}\) → Sides are 15, \(15\sqrt{3}\), and 30. Sum them up: \(15 + 30 + 15\sqrt{3} = 45 + 15\sqrt{3}\). 10. \(108 + 36\sqrt{3}\) → Hypotenuse = 72, short leg = 36, long leg \(= 36\sqrt{3}\). Sum: \(72 + 36 + 36\sqrt{3} = 108 + 36\sqrt{3}\).