A triangle has sides measuring 5 cm, 12 cm, and 13 cm. Determine if it is a right triangle.
["Title: Is a Triangle with Sides 5 cm, 12 cm, and 13 cm a Right Triangle? A Scientific Exploration", "When analyzing geometric shapes, one of the most fundamental questions is whether a triangle is a right triangle—a triangle with one 90-degree angle. A classic example often studied in geometry classrooms and engineering applications includes a triangle with side lengths of 5 cm, 12 cm, and 13 cm. Is this truly a right triangle? Let’s determine this with clarity and precision.", "### Understanding Right Triangles", "A right triangle is defined by the Pythagorean Theorem: in a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. Mathematically, if a triangle has sides ( a ), ( b ), and ( c ) (where ( c ) is the hypotenuse), then:", "[\na^2 + b^2 = c^2\n]", "This principle is foundational in geometry and widely used in physics, architecture, and computer graphics.", "### Applying the Pythagorean Theorem to 5 cm, 12 cm, and 13 cm", "To verify if the triangle with sides 5 cm, 12 cm, and 13 cm is right-angled, we follow these steps:", "1. Identify the Hypotenuse\n Since these are positive lengths, assume the longest side, 13 cm, is the hypotenuse.", "2. Calculate ( a^2 + b^2 )\n Let ( a = 5 ), ( b = 12 ). Then:", "[\n a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169\n ]", "3. Calculate ( c^2 )\n [\n c^2 = 13^2 = 169\n ]", "4. Compare Both Sides", "[\n a^2 + b^2 = 169 = c^2\n ]", "### Conclusion: It Is a Right Triangle", "Since ( 5^2 + 12^2 = 13^2 ), the triangle satisfies the Pythagorean Theorem. Therefore, a triangle with side lengths 5 cm, 12 cm, and 13 cm is indeed a right triangle—with the hypotenuse measuring 13 cm.", "### Why This Matters", "Recognizing right triangles simplifies calculations involving distances, angles, and structural stability. For instance, 5–12–13 triangles appear frequently because they are a scaled version of the smallest integer right triangle (3–4–5). This makes them valuable in construction, navigation, and design.", "In summary: Yes, a triangle with sides 5 cm, 12 cm, and 13 cm is a right triangle, confirmed through the Pythagorean Theorem. Understanding such properties empowers problem-solving across science, technology, and everyday applications.", "---", "Keywords: right triangle, 5 cm triangle, 12 cm triangle, 13 cm triangle, Pythagorean Theorem, right triangle verification, geometry principles, right triangle 5-12-13, math theorem explanation, geometric shapes.\nMeta Description: A triangle with sides 5 cm, 12 cm, and 13 cm is a right triangle because it satisfies the Pythagorean Theorem: ( 5^2 + 12^2 = 13^2 ). Learn how to verify right triangles using geometry."]
