A ladder 13 meters long leans against a wall, reaching a point 12 meters up. If the base is moved 2 meters farther from the wall, how high does it reach?

Thu, 07 May 2026 19:06:08 +0000

**A ladder 13 meters long leans against a wall, reaching a point 12 meters up. If the base is moved 2 meters farther from the wall, how high does it reach?** When a 13-meter ladder leans against a wall and touches 12 meters high, it forms a clear geometric relationship rooted in the Pythagorean theorem—where the ladder is the hypotenuse, the wall position is one vertical leg, and the distance from the wall forms the other. This practical scenario isn’t just a static image; it’s a real-world question gaining subtle traction, especially among homeowners, DIY enthusiasts, and DIY contractors seeking smart space optimization. ### The Geometry Behind the Leaning Ladder
With a 13-meter ladder leaning at a right angle, the vertical reach depends directly on how far its base stays from the wall. When the base stands 5 meters from the wall—calculated via √(13² – 12²) = √169 – 144 = √25 = 5—the ladder reaches exactly 12 meters. Now, shifting the base 2 meters farther increases that distance to 7 meters. Using the same formula, we find the new height by computing √(13² – 7²) = √(169 – 49) = √120 ≈ 10.95 meters. The ladder climbs higher—since moving it away increases vertical reach—though not to full height. ### Why This Math Matters in Modern Living
In cities nationwide, space constraints are prompting creative solutions. Rooftop access, balcony repairs, and garden extensions rely on precise leverage of vertical reach. The ripple effect of this simple adjustment—changing from 5m to 7m base distance—shows how a small shift optimizes safety and efficiency, reducing strain and risk. For professionals and DIYers alike, understanding this dynamic highlights the value of planning before deployment.