Hypotenuse: \( \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \) cm.
["# Understanding the Hypotenuse: Calculating ( \sqrt{9^2 + 12^2} = 15 ) cm", "When learning about right triangles, one of the most essential formulas is the Pythagorean Theorem:\n[\nc = \sqrt{a^2 + b^2}\n]\nwhere ( c ) represents the length of the hypotenuse, and ( a ) and ( b ) are the lengths of the other two sides.", "### What is the Hypotenuse?", "The hypotenuse is always the longest side of a right-angled triangle, opposite the right angle. Its length depends on the lengths of the two legs, and calculating it precisely ensures accurate design, construction, or analysis in fields like architecture, engineering, and physics.", "---", "## Solving: Find the Hypotenuse of a Triangle with Sides 9 cm and 12 cm", "Given the two legs:\n- ( a = 9 ) cm\n- ( b = 12 ) cm", "Apply the Pythagorean Theorem:\n[\nc = \sqrt{a^2 + b^2} = \sqrt{9^2 + 12^2}\n]", "### Step-by-Step Calculation:", "1. Compute the squares:\n[\n9^2 = 81 \quad \ ext{and} \quad 12^2 = 144\n]\n2. Add the results:\n[\n81 + 144 = 225\n]\n3. Take the square root:\n[\nc = \sqrt{225} = 15\n]", "---", "## Final Result", "[\n\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \ ext{ cm}\n]", "The hypotenuse measures 15 centimeters.", "---", "## Why This Formula Matters", "Determining the hypotenuse accurately is crucial in many real-world applications—from calculating ladder stability to designing building structures, designing electronics, or even in GPS navigation. Using the Pythagorean Theorem ensures precision in measurements, reducing errors and optimizing safety and efficiency.", "---", "### Quick Recap", "- Right triangle hypotenuse formula: ( c = \sqrt{a^2 + b^2} )\n- For a 9 cm and 12 cm right triangle:\n[\nc = \sqrt{9^2 + 12^2} = 15 \ ext{ cm}\n]\n- Perfect for teaching, engineering, and everyday math problems.", "---", "Keywords: hypotenuse calculation, Pythagorean Theorem, square root of squares, geometry, right triangle formula, ( \sqrt{81 + 144} = 15 ), geometry tutorial, physics applications, architectural calculation, educational math."]
