A ladder 13 meters long leans against a wall, reaching a height of 12 meters. How far is the base of the ladder from the wall?
["How Far Is the Base of a 13-Meter Ladder From the Wall When It Reaches 12 Meters High?", "When dealing with basic ladder safety and geometry, one question often pops up: If a 13-meter ladder leans against a wall and reaches a height of 12 meters, how far is the base of the ladder from the wall? This scenario beautifully illustrates the application of the Pythagorean theorem — a fundamental principle in geometry that holds true regardless of real-world applications like ladder placement.", "### The Geometry Behind the Problem", "Imagine a vertical ladder standing against a vertical wall. The ladder forms the hypotenuse of a right triangle — the wall acts as one leg (vertical), and the distance from the base of the ladder to the wall is the other leg (horizontal). The height the ladder reaches on the wall is the opposite side relative to the base.", "We are given:\n- The ladder’s full length (hypotenuse (c)) = 13 meters\n- The height reached on the wall (opposite side) = 12 meters\n- The distance from the wall (adjacent side) is unknown and must be calculated.", "This fits perfectly with the Pythagorean theorem:", "[\na^2 + b^2 = c^2\n]", "Where:\n- (a) = distance from the wall (what we’re solving for)\n- (b = 12) meters\n- (c = 13) meters", "### Applying the Values", "Plug in the known values:", "[\na^2 + 12^2 = 13^2\n]\n[\na^2 + 144 = 169\n]\n[\na^2 = 169 - 144 = 25\n]\n[\na = \sqrt{25} = 5\n]", "### Conclusion", "The base of the 13-meter ladder is 5 meters from the wall when it reaches a height of 12 meters. This calculation ensures the ladder maintains safe positioning while maximizing reach — a classic example of how geometry keeps us safe in construction, DIY, and daily life.", "### Extra Tips for Ladder Safety", "- Always check the ladder’s specification — a 13-meter ladder is ideal for reaching 12 meters safely without overreaching.\n- The base should stay no more than 6 meters from the wall for optimal stability (based on a 5-12-13 ratio and safe ergonomic distances).\n- Inspect the ladder surface and joints before use.\n- Use a stable, level surface.", "Understanding this simple geometric relationship empowers you to apply safety and precision in any stairway, climbing, or support task. Stay safe and stay informed!", "---", "Keywords: ladder distance from wall, 13 meter ladder at 12 meters height, math ladder problem, Pythagorean theorem, ladder safety, how far from wall, right triangle ladder calculation"]
