A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?
["Finding the Hypotenuse of a Right Triangle with Legs 9 and 12: A Step-by-Step Guide", "Understanding the fundamental properties of right triangles is essential for solving countless problems in geometry, trigonometry, and real-world applications. One of the most common questions students and enthusiasts encounter is: What is the length of the hypotenuse in a right triangle with legs of length 9 and 12?", "In this article, we’ll walk through the process using the Pythagorean Theorem, a foundational principle that applies to all right triangles. Whether you're studying for a math test, tackling a homework problem, or simply curious about triangle geometry, this guide will clarify how to calculate the hypotenuse accurately.", "### What is a Right Triangle?", "A right triangle is a triangle with one angle exactly equal to 90 degrees. The two sides forming this right angle are called the legs, and the opposite side—the longest side—is the hypotenuse.", "### Applying the Pythagorean Theorem", "The Pythagorean Theorem states:", "[\na^2 + b^2 = c^2\n]", "Where:\n- ( a ) and ( b ) are the lengths of the legs,\n- ( c ) is the length of the hypotenuse.", "In this problem, the leg lengths are given as:\n- ( a = 9 )\n- ( b = 12 )", "Substituting into the formula:", "[\n9^2 + 12^2 = c^2\n]", "[\n81 + 144 = c^2\n]", "[\n225 = c^2\n]", "To find ( c ), take the square root of both sides:", "[\nc = \sqrt{225} = 15\n]", "### Final Answer", "The length of the hypotenuse of a right triangle with legs 9 and 12 is 15 units.", "### Practical Applications", "Knowing how to calculate the hypotenuse is not only academically valuable—it has real-world significance. This concept plays a key role in:", "- Architecture and construction, where right angles and diagonal measurements ensure structural integrity\n- Navigation and surveying, where distances and angles need precise computation\n- Pixels and graphics, where diagonal screen lengths depend on triangle relationships", "### Summary", "To summarize an efficient solution:", "1. Identify the legs (9 and 12).\n2. Apply ( c = \sqrt{a^2 + b^2} ).\n3. Compute: ( c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 ).\n4. The hypotenuse is 15.", "Mastering the Pythagorean Theorem equips you with a powerful tool for solving triangle-based problems. So next time you're asked, What is the hypotenuse?—know the answer is 15 for a triangle with legs 9 and 12.", "---", "Keywords: right triangle, hypotenuse, Pythagorean Theorem, legs 9 and 12, triangle geometry, math tutorial, problem solving, geometry basics"]
