A ladder leans against a wall, reaching a height of 12 meters. If the base of the ladder is 5 meters from the wall, what is the length of the ladder?
["Title: How to Calculate the Length of a Ladder Reaching 12 Meters on a Wall – A Step-by-Step Guide", "When you see a sturdy ladder leaning against a wall, reaching impressive heights, you might wonder: How long is the ladder? This is a classic problem in geometry—specifically involving right triangles—common in construction, home improvement, and physics. In this article, we’ll explore how to calculate the length of a ladder that reaches a height of 12 meters while maintaining a base 5 meters away from the wall. Plus, we’ll explain the math behind it using the Pythagorean theorem.", "---", "### Understanding the Ladder Problem", "Imagine a ladder standing vertically against a wall—forming a right triangle with the ground.\n- The height the ladder reaches on the wall = 12 meters (one leg of the triangle).\n- The distance from the wall’s base to under the ladder = 5 meters (the other leg).\n- The ladder itself is the hypotenuse—the longest side—connecting the top of the ladder to the base on the ground.", "This setup follows the Pythagorean theorem, a foundational principle in geometry:", "[\na^2 + b^2 = c^2\n]", "Where:\n- ( a = 5 ) meters (base)\n- ( b = 12 ) meters (height)\n- ( c = \ ext{ladder length (what we are solving for)} )", "---", "### Step-by-Step Calculation", "1. Write down the equation:\n[\nc^2 = a^2 + b^2\n]", "2. Substitute known values:\n[\nc^2 = 5^2 + 12^2 = 25 + 144 = 169\n]", "3. Take the square root to find ( c ):\n[\nc = \sqrt{169} = 13\n]", "---", "### ✅ Final Answer: The ladder is 13 meters long", "This elegant result—known as the 13-12-5 Pythagorean triple—is a perfect example of how math brings real-world problems into clear focus. Regardless of whether the ladder reaches 12 meters with a 5-meter base or other heights, knowing the geometric relationship helps ensure safety, precision, and optimal use of materials.", "---", "### Practical Takeaways", "- Use the Pythagorean theorem whenever you need to find the length of a diagonal in a right-angled triangle.\n- This calculation applies not just to ladders, but also to roof trusses, diagonal brakes in cars, and stair construction.\n- Always prioritize ladder safety—ensure it’s placed 3 meters away from the wall for every 4 meters up the wall (the 4:3 rule) to reduce tipping risks.", "---", "### Summary", "- Height: 12 meters\n- Base distance: 5 meters\n- Ladder length (hypotenuse): 13 meters", "Understanding this simple yet powerful calculation ensures confidence and accuracy in any task involving right triangles—whether you’re climbing a ladder or designing structures.", "---", "Meta keywords: ladder length calculation, Pythagorean theorem, ladder safety, right triangle geometry, how long is a 12m ladder, ladder 5m from wall", "FAQ:\nQ: What ladder length reaches 12 meters at a 5-meter base?\nA: Exactly 13 meters, computed using ( \sqrt{5^2 + 12^2} = 13 ).", "Q: Why is 5-12-13 a Pythagorean triple?\nA: Because ( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 ), satisfying the right triangle Pythagorean condition."]
