A triangle has sides in the ratio 3:4:5. If the perimeter is 60 cm, find the area.
["Finding the Area of a 3:4:5 Triangle with a Perimeter of 60 cm", "Triangles with sides in the ratio 3:4:5 are special right triangles known for forming a perfect Pythagorean triple—commonly used in geometry to simplify area and perimeter calculations. In this article, we explore how to find the area of such a triangle when its perimeter measures exactly 60 cm.", "### Understanding the 3:4:5 Triangle Ratio", "The sides of the triangle are proportional to 3, 4, and 5. Let the actual side lengths be:", "- (3x)\n- (4x)\n- (5x)", "This triangle is a right triangle, where the 5x side serves as the hypotenuse, satisfying the Pythagorean theorem:\n[\n(3x)^2 + (4x)^2 = (5x)^2 \Rightarrow 9x^2 + 16x^2 = 25x^2\n]\nwhich verifies the 3-4-5 ratio as a right triangle.", "### Using the Perimeter to Find the Scale Factor (x)", "The perimeter of the triangle is the sum of its sides:\n[\n3x + 4x + 5x = 12x\n]\nGiven that the perimeter equals 60 cm:\n[\n12x = 60 \Rightarrow x = 5\n]", "### Calculating the Actual Side Lengths", "Multiplying each ratio part by (x = 5):\n- Leg 1: (3x = 3 \ imes 5 = 15) cm\n- Leg 2: (4x = 4 \ imes 5 = 20) cm\n- Hypotenuse: (5x = 25) cm", "### Calculating the Area of the Triangle", "Since this is a right triangle, the area (A) is calculated using the two perpendicular legs:\n[\nA = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} = \frac{1}{2} \ imes 15 \ imes 20 = \frac{1}{2} \ imes 300 = 150 \ ext{ cm}^2\n]", "### Summary", "- Sides: 15 cm, 20 cm, 25 cm\n- Perimeter: 60 cm\n- Area: 150 cm²", "This triangle exemplifies how proportional reasoning and Pythagorean triples simplify geometric calculations. Whether for academic work, construction, or design, understanding how ratios and perimeters relate allows quick and accurate area determination.", "---", "Key SEO Keywords: 3:4:5 triangle, triangle area calculation, right triangle perimeter, perimeter 60 cm triangle, 3-4-5 triangle area, find area of 3-4-5 triangle", "---", "Use this method whenever working with 3:4:5 triangles—your geometry questions will always be solved efficiently and accurately."]
