A triangle has sides in the ratio 3:4:5. If the perimeter is 60 cm, what are the lengths of the sides?
["Understanding Triangle Side Ratios: The Classic 3:4:5 Ratio and Its Real-World Application", "Triangles are foundational shapes in geometry, with countless real-world applications ranging from architecture to engineering. One classic and widely recognized triangle is the 3:4:5 triangle, known for its unique properties and practical usability—especially in construction and design. In this article, we explore how to calculate the actual side lengths when the triangle’s perimeter is given, using the famous 3:4:5 side ratio.", "---", "### What is a 3:4:5 Triangle?", "The 3:4:5 triangle is a right-angled triangle, satisfying the Pythagorean theorem: (3^2 + 4^2 = 5^2) (9 + 16 = 25). This means it has a right angle between the sides of lengths 3 and 4, with the hypotenuse measuring 5. This ratio is particularly valuable because it simplifies many geometric problems and ensures a perfect right angle.", "---", "### Why Is the 3:4:5 Ratio Important?", "This triangle is not just mathematically elegant—it’s practical. Builders and engineers often use this ratio for framing and alignment because it guarantees accurate right angles without measuring angles directly.", "---", "### Given Problem: Perimeter of 60 cm", "Let’s solve a common application problem:\nIf a triangle has side lengths in the ratio 3:4:5 and a perimeter of 60 cm, what are the actual lengths of its sides?", "---", "### Step-by-Step Solution", "Let the sides of the triangle be:\n- (3x) cm (shortest side)\n- (4x) cm (middle side)\n- (5x) cm (hypotenuse)", "Since the perimeter is the sum of all sides:\n[\n3x + 4x + 5x = 60\n]", "Combine like terms:\n[\n12x = 60\n]", "Solve for (x):\n[\nx = \frac{60}{12} = 5\n]", "Now calculate each side:\n- First side: (3x = 3 \ imes 5 = 15) cm\n- Second side: (4x = 4 \ imes 5 = 20) cm\n- Third side: (5x = 5 \ imes 5 = 25) cm", "---", "### Summary of Side Lengths", "| Side | Length (cm) |\n|------|-------------|\n| Side a | 15 cm |\n| Side b | 20 cm |\n| Hypotenuse | 25 cm |", "---", "### Final Checks", "Verify the perimeter:\n[\n15 + 20 + 25 = 60 \ ext{ cm} \quad \ ext{✅ Valid}\n]\nCheck the Pythagorean theorem:\n[\n15^2 + 20^2 = 225 + 400 = 625 = 25^2 \quad \ ext{✅ Right-angled}\n]", "---", "### Conclusion", "The 3:4:5 triangle with a perimeter of 60 cm has side lengths of 15 cm, 20 cm, and 25 cm. Combining proportional reasoning with algebraic calculation, we solved this classic geometry problem efficiently. Whether you're a student, learner, or peidity enthusiast, mastering such ratios strengthens your foundation in geometry and real-world problem solving.", "If you're studying triangles or planning a DIY project involving precise angles, remember: the 3:4:5 triangle, scaled smartly, is your go-to shape—simple, strong, and accurate.", "---", "Keywords: 3:4:5 triangle, triangle side lengths, perimeter calculation, right triangle math, Pythagorean theorem, geometry tutorial, practical triangle applications", "Meta Description:\nDiscover how to calculate the side lengths of a 3:4:5 triangle when the perimeter is 60 cm. Step-by-step solution with verification and real-world relevance—perfect for students and DIY enthusiasts."]
