To find the hypotenuse \(c\) of a right triangle with legs \(a = 9\) and \(b = 12\), we use the Pythagorean theorem:

["How to Find the Hypotenuse ( c ) of a Right Triangle Using the Pythagorean Theorem", "When solving for the hypotenuse ( c ) of a right triangle, the Pythagorean theorem is the most reliable and widely used mathematical principle. This fundamental rule applies only to right triangles—triangles with one 90-degree angle—and relates the three sides as follows:", "[\nc^2 = a^2 + b^2\n]", "where ( c ) is the hypotenuse (the side opposite the right angle), and ( a ) and ( b ) are the lengths of the other two legs.", "### Step-by-Step: Finding the Hypotenuse with ( a = 9 ) and ( b = 12 )", "Let’s apply this formula to find the hypotenuse when legs ( a = 9 ) and ( b = 12 ).", "1. Write the Pythagorean equation:\n[\nc^2 = a^2 + b^2\n]", "2. Substitute the known values:\n[\nc^2 = 9^2 + 12^2\n]", "3. Calculate the squares of the legs:\n[\n9^2 = 81,\quad 12^2 = 144\n]", "4. Add the results:\n[\nc^2 = 81 + 144 = 225\n]", "5. Solve for ( c ) by taking the square root:\n[\nc = \sqrt{225} = 15\n]", "### Conclusion", "The hypotenuse of the right triangle with legs ( a = 9 ) and ( b = 12 ) is:", "[\n\boxed{15}\n]", "Using the Pythagorean theorem not only gives an accurate result quickly but also strengthens foundational math skills essential in geometry, trigonometry, and real-world applications like construction, navigation, and physics. Whether you're a student, teacher, or problem solver, mastering this simple formula opens the door to solving countless triangle-related challenges."]