By the Pythagorean theorem: \( 9^2 + 15^2 = c^2 \).

["Understanding the Pythagorean Theorem: Solving ( 9^2 + 15^2 = c^2 )", "The Pythagorean theorem is one of the most fundamental principles in geometry, serving as a cornerstone for understanding right triangles and distance calculations. Derived from ancient mathematics, this theorem reveals a powerful relationship between the sides of a right-angled triangle: the square of the hypotenuse equals the sum of the squares of the other two sides.", "A classic example often used to illustrate this concept is the equation:\n[ 9^2 + 15^2 = c^2 ]", "Let’s break this down step by step to apply the Pythagorean theorem and solve for ( c ), the hypotenuse.", "---", "### What Is the Pythagorean Theorem?", "At its core, the theorem states that in a right triangle with legs ( a ) and ( b ), and hypotenuse ( c ):\n[\na^2 + b^2 = c^2\n]", "This formula is essential not only in classical geometry but also in fields like physics, engineering, computer graphics, and navigation.", "---", "### Applying the Theorem to ( 9^2 + 15^2 = c^2 )", "Given:\n[\na = 9, \quad b = 15\n]", "We substitute into the equation:\n[\nc^2 = 9^2 + 15^2\n]", "Now calculate each square:\n[\n9^2 = 81\n]\n[\n15^2 = 225\n]", "Add them together:\n[\nc^2 = 81 + 225 = 306\n]", "To find ( c ), take the square root of both sides:\n[\nc = \sqrt{306}\n]", "---", "### Simplifying the Square Root", "While ( \sqrt{306} ) is an exact answer, we can simplify it further by factoring:\n[\n306 = 9 \ imes 34\n]\nSo,\n[\n\sqrt{306} = \sqrt{9 \ imes 34} = \sqrt{9} \cdot \sqrt{34} = 3\sqrt{34}\n]", "Thus, the precise length of the hypotenuse is:\n[\nc = 3\sqrt{34}\n]", "For exact measurements, the irrational form ( 3\sqrt{34} ) is preferred. Alternatively, as a decimal, ( \sqrt{306} \approx 17.49 ), but the symbolic form preserves mathematical accuracy.", "---", "### Real-World Applications of This Triangle", "Triangles whose sides relate via ( 9:15:c ) are not just abstract—they appear frequently in practical scenarios. For example:", "- Construction: Used to verify right angles in building foundations or roof trusses.\n- Surveying: To calculate diagonal distances between two points.\n- Physics: Resolving vectors into perpendicular components.", "Knowing that the hypotenuse measures ( 3\sqrt{34} ) allows professionals to work with exact or approximate values depending on accuracy needs.", "---", "### Final Thoughts", "The Pythagorean theorem remains a timeless mathematical tool. By solving ( 9^2 + 15^2 = c^2 ), we find:\n[\nc = \sqrt{306} = 3\sqrt{34}\n]\na clean, exact expression that supports both theoretical understanding and real-world applications.", "Whether you're a student learning geometry basics or a professional applying trigonometric principles, mastering this theorem and its computation opens doors to deeper mathematical insight.", "---", "Keywords: Pythagorean theorem, right triangle calculation, ( 9^2 + 15^2 = c^2 ), hypotenuse length, ( c = 3\sqrt{34} ), geometry, trigonometry, mathematical applications.", "---", "Explore more about geometry and theorems – mastering the Pythagorean principle empowers precise calculations in countless practical and academic contexts."]