A ladder leans against a wall, forming a right triangle with the ground. The ladder is 13 meters long, and the base is 5 meters from the wall. If the ladder slides so the base moves 2 meters farther out, how far does the top of the ladder slide down the wall?
["How a 13-Meter Ladder Sliding Against a Wall Affects Its Height: A Right Triangle Problem Explained", "When a sturdy 13-meter ladder leans against a wall, forming a right triangle with the ground, it creates a compelling geometric scenario that turns into a real-life physics problem. Understanding how the ladder’s position changes involves basic trigonometry and geometry — particularly when the base shifts and the ladder slides down.", "### The Initial Setup", "Let’s start with the initial position:", "- The ladder (hypotenuse) is fixed at 13 meters.\n- The base is 5 meters from the wall.\n- Use the Pythagorean theorem to find the initial height on the wall.", "[\n\ ext{Height} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \ ext{ meters}\n]", "So, the top of the ladder reaches 12 meters high initially.", "### After the Base Slides", "The base moves 2 meters farther out, so it’s now 7 meters from the wall. The ladder remains 13 meters long, forming a new right triangle. Let’s calculate the new height.", "Using the Pythagorean theorem again:", "[\n\ ext{New height} = \sqrt{13^2 - 7^2} = \sqrt{169 - 49} = \sqrt{120} \approx 10.95 \ ext{ meters}\n]", "### How Far the Top Slides Down", "Subtract the new height from the original:", "[\n12,\ ext{m} - \sqrt{120} \approx 12 - 10.95 = 1.05,\ ext{meters}\n]", "So, the top of the ladder slides down approximately 1.05 meters as the base moves 2 meters farther from the wall.", "### Visualizing the Change", "This problem beautifully illustrates how changing one side in a right triangle affects the others. As the base increases, the height decreases, maintaining the 13-meter ladder length — a perfect example of the Pythagorean theorem in action.", "---", "Summary:\n- Initial height: 12 meters\n- After base slides 2 meters (now at 7 meters): ~10.95 meters\n- The top slides down approximately 1.05 meters", "This simple yet precise scenario demonstrates how geometry underpins real-world stability and motion — ideal for both math students and anyone curious about how objects behave under change.", "---", "Keywords: ladder leaning against wall, right triangle ladder problem, Pythagorean theorem, ladder sliding down wall, geometry in real life, slope of ladder, ladder height change, 13-meter ladder geometry", "Meta description: Discover how a 13-meter ladder leaning 5 meters from a wall reaches 12 meters high, then slides down approximately 1.05 meters when the base moves 2 meters away. See the math behind the slide."]
