A ladder leans against a wall, forming a right triangle with the ground. If the ladder is 13 feet long and reaches 12 feet up the wall, how far is its base from the wall?

["## How to Solve the Classic Right Triangle Problem: The Ladder Against the Wall", "When visualizing a ladder leaning against a wall, a simple yet powerful geometry concept comes to life—a right triangle formed by the ladder, the wall, and the ground. Understanding this setup helps in solving real-world problems involving measurements, angles, and stability.", "The Scenario in Detail\nA 13-foot-long ladder leans firmly against a wall, reaching 12 feet up. To find out how far the base of the ladder is from the wall, we can apply the Pythagorean theorem—a fundamental principle in geometry.", "The Math Behind It\nIn a right triangle:\n- The hypotenuse is the longest side, opposite the right angle. Here, the ladder (13 feet) is the hypotenuse.\n- One leg is vertical (the height climbed up the wall: 12 feet).\n- The other leg is horizontal—the distance from the wall’s base to the ladder’s bottom (what we want to find).", "According to the Pythagorean theorem:\n[\n\ ext{Hypotenuse}^2 = \ ext{Leg}_1^2 + \ ext{Leg}_2^2\n]\nSubstitute the known values:\n[\n13^2 = 12^2 + x^2\n]\n[\n169 = 144 + x^2\n]\n[\nx^2 = 169 - 144 = 25\n]\n[\nx = \sqrt{25} = 5\n]", "Conclusion\nThe base of the ladder is exactly 5 feet away from the wall.", "Real-World Application\nThis triangle model isn’t just academic—it helps builders, painters, and homeowners estimate ratios, ensure safe placement, and prevent accidents. Whether securing a ladder or designing structures, right triangles provide reliable, efficient solutions.", "Understanding these principles empowers you to approach similar problems with confidence and precision. Next time a ladder leans against a wall, recall the 5-12-13 triangle—and trust geometry to guide your safety and measurements."]