A 15-meter ladder leans against a wall, reaching a height of 12 meters. How far is the base from the wall?
["Understanding the Geometry Behind a 15-Meter Ladder Reaching 12 Meters High: Where Should the Base Stand?", "When dealing with ladders, safety is paramount—and so is understanding the math behind proper ladder placement. A common real-world example involves a 15-meter-long ladder leaning against a wall, yet reaching only 12 meters high. This situation raises an essential geometric question: How far from the wall should the base of the ladder be placed?", "### The Ladder Model: A Right Triangle in Action", "Imagine a ladder leaning against a vertical wall forms a right triangle with the ground and the wall. The ladder acts as the hypotenuse, measuring 15 meters. The vertical height it reaches on the wall is 12 meters, and the horizontal distance from the wall base to the foot forms the base leg of the triangle.", "### Applying the Pythagorean Theorem", "To find the distance from the ladder base to the wall, we use the Pythagorean theorem, which states:", "[\n\ ext{hypotenuse}^2 = \ ext{base}^2 + \ ext{height}^2\n]", "Let’s plug in the known values:", "- Hypotenuse (ladder length) = 15 m\n- Height (vertical reach) = 12 m", "[\n15^2 = \ ext{base}^2 + 12^2\n]", "[\n225 = \ ext{base}^2 + 144\n]", "[\n\ ext{base}^2 = 225 - 144 = 81\n]", "[\n\ ext{base} = \sqrt{81} = 9\n]", "### Conclusion: The Base Should Be 9 Meters from the Wall", "So, if your 15-meter ladder reaches 12 meters high on the wall, the base must be positioned 9 meters away from the wall to maintain safe and correct positioning.", "This simple application of geometry ensures both effectiveness and stability—keeping climbers safe and equipment functioning properly. Whether you’re a homeowner, a contractor, or simply curious about ladder safety, remembering the Pythagorean triple (9–12–15) can save time, minimize risk, and reinforce sound math habits.", "---", "Key Takeaways:\n- A 15-meter ladder with 12-meter height forms a right triangle.\n- Use (c^2 = a^2 + b^2) to calculate distance.\n- The base should be exactly 9 meters from the wall.\n- Prioritize accurate placement for safety and stability.", "---", "Keywords: ladder placement calculator, how far should ladder base be from wall, 15-meter ladder height calculation, right triangle ladder problem, ladder safety geometry, Pythagorean theorem ladder answer, 9 meter ladder distance guide."]
