📐 勾股定理：直角三角形的秘密公式📐 Pythagorean Theorem: the Right Triangle's Secret Formula
上一片讲圆的性质时发现：半圆里藏着一个直角。🧮 小算盘机器人对这个直角特别感兴趣——它的三条边之间锁着一个流传两千多年的公式，今天就来解开它。Last leaf we found a right angle hiding in every semicircle. The little calculator robot is fascinated: that right angle locks a two-thousand-year-old formula between its three sides. Time to unlock it.
第 1 步：三个正方形的拼图Step 1: A puzzle of three squares
先从一幅画看起——直角三角形的三条边上，各长出一个正方形。Start with a picture: a square grows on each side of a right triangle.
直角三角形的两条短边叫直角边（a 和 b），最长的那条叫斜边（c）。沿每条边向外画一个正方形：两个小正方形的面积是 a² 和 b²，大正方形的面积是 c²。神奇的是，两个小正方形的面积加起来，正好把大正方形填得满满当当——这就是 a² + b² = c²，对所有直角三角形都成立。A right triangle's two shorter sides are the legs (a and b); the longest side is the hypotenuse (c). Build a square on each side: the small squares have areas a² and b², the big one c². Wonderfully, the two small areas add up to exactly fill the big square — that is a² + b² = c², true for every right triangle.
公式有了，可实际量直角时，取哪几个数字最省事？The formula works — but which numbers make measuring a right angle easiest?
第 2 步：3-4-5，工人的直角神器Step 2: 3-4-5, the builder's right-angle trick
取直角边 3 和 4，斜边正好是 5：9 + 16 = 25。像这样三条边都是整数的组合叫“勾股数”，3-4-5 是最小的一组。古埃及的拉绳工人把绳子 12 等分打结，围出 3 段、4 段、5 段——直角立刻出现，不用量角器。把 3-4-5 同乘一个数（比如 6-8-10），公式依然成立。Take legs 3 and 4 — the hypotenuse comes out exactly 5: 9 + 16 = 25. Whole-number trios like this are Pythagorean triples, and 3-4-5 is the smallest. Ancient rope stretchers knotted a rope into 12 equal parts and folded out sides of 3, 4 and 5 — a perfect right angle, no protractor needed. Scale the triple by any number (6-8-10) and it still works.
知道两条边，就能算出第三条——工地和生活中谁在用？Two sides known, the third follows — who uses this on the job?
第 3 步：梯子靠墙，随手一用Step 3: Ladder against the wall
梯子、墙和地面围成一个直角三角形。墙高 4 米、梯脚离墙 3 米，梯子至少要多长？4² + 3² = 25，所以梯长 5 米。反过来也行：知道梯长和离墙的距离，就能算出够得着多高。电视屏幕对角线、两点间的直线距离、盖房子放线找直角，用的全是这一个公式。Ladder, wall and ground form a right triangle. Wall 4 m high, ladder foot 3 m from the wall — how long must the ladder be? 4² + 3² = 25, so 5 m. The reverse works too: from the ladder's length and its distance from the wall you get the reachable height. Screen diagonals, straight-line distances, squaring up a building — all the same formula.
🎮 你来拼正方形（30 秒）🎮 Your turn: tile the squares (30 seconds)
道理讲完了。拖动两条直角边 a、b，看两个小正方形的面积实时流进斜边大正方形；把 a² + b² 调成目标值，连过 4 关。Theory done. Drag the legs a and b and watch the two small squares flow into the big one; tune a² + b² to the target and clear 4 levels.
一句话记住它：直角三角形里，两条直角边的平方和 = 斜边的平方（a² + b² = c²）。Remember it in one line: in a right triangle, the squares of the two legs add up to the square of the hypotenuse (a² + b² = c²).
直角边上的两个正方形面积，正好填满斜边正方形The two leg-squares exactly fill the hypotenuse square3-4-5 是最顺手的勾股数，同乘一个数依然成立3-4-5 is the handiest triple; scaling it keeps it true知二求三：梯子、距离、对角线，都是同一个公式Know two sides → the third follows: ladders, distances, diagonals
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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