A ladder 13 meters long leans against a wall, reaching height 12 meters. How far is the base from the wall?
["How Far Is the Base of a 13-Meter Ladder From the Wall When It Reaches 12 Meters High?", "Ever wondered how a 13-meter ladder can safely reach 12 meters up a wall without slipping? This common question involves geometry and the Pythagorean theorem — and the answer is both practical and intriguing. In this article, we’ll explore the math behind a 13-meter ladder leaning against a wall at 12 meters high, and how far its base sits from the wall.", "---", "### Understanding the Problem: The Ladder as a Right Triangle", "When a ladder leans against a wall, it forms a right triangle with the ground and the vertical wall. The ladder serves as the hypotenuse — the longest side — measuring 13 meters. The wall height represents one leg of the triangle at 12 meters, and the distance from the base of the ladder to the wall is the other leg, which we are solving for.", "---", "### Using the Pythagorean Theorem", "The Pythagorean theorem states:\n( a^2 + b^2 = c^2 )\nWhere:\n- ( c ) = length of the hypotenuse (ladder) = 13 meters\n- ( b ) = height reached on the wall = 12 meters\n- ( a ) = distance from base of ladder to wall (what we want to find)", "Plugging in known values:\n[\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 \ ext{ meters}\n]", "---", "### Final Result: The Base of the Ladder Is 5 Meters from the Wall", "So, when a 13-meter ladder leans against a wall and reaches a height of 12 meters, the base of the ladder rests 5 meters from the wall. This positioning ensures the ladder stays stable and in contact with both the wall and the ground.", "---", "### Practical Implications and Tips", "Knowing this distance helps ensure safety and stability when setting up a ladder:\n- A 5 m base-to-wall distance pairs well with a 12 m height for optimal balance.\n- Avoid placing ladders too close to the wall, as this increases the risk of slipping.\n- When leaning against walls, always face the ladder directly, keeping it steady and parallel to the ground.", "---", "### Bonus: Why It’s More Than Just Math", "Beyond equations, this problem illustrates how geometry applies to real-life safety — ensuring ladders are used correctly prevents accidents in homes, construction sites, and workplaces worldwide.", "---", "Summary\nA 13-meter ladder leaning at 12 meters high rests its base 5 meters from the wall, forming a classic 5-12-13 right triangle. This ratio is a trusted benchmark for safely using ladders in daily tasks.", "---", "Keywords: ladder leaning against wall, 13 meter ladder height 12 meters, math problem ladder geometry, safety ladder distance, right triangle ladder height, how far from wall 13m ladder, Pythagorean theorem ladder, ladder placement guide\nMeta Description: Learn how the base of a 13-meter ladder reaches 12 meters high by applying the Pythagorean theorem — distance is 5 meters from the wall. Ideal for ladder safety and practical math.\nISC Articles: #LadderSafety #GeometryApplications #HomeDIYMath #RightTriangleProblems"]
