📐 坐标与距离：两点之间有多远📐 Coordinates & Distance: How Far Between Two Points?
上一片我们从三个方向看一个立体图形。这一片换个问题：方格城市里，小算盘机器人要从 A 点走到 B 点，两点之间到底有多远？Last leaf we looked at a solid from three directions. This leaf asks a different question: in Grid City, the little abacus robot walks from point A to point B — exactly how far apart are they?
第 1 步：横平竖直，直接减Step 1: Flat or upright — just subtract
斜线不好直接量——先把横、竖两个方向的距离量出来。A slanted line is awkward to measure — start with the two straight directions.
在方格城市里，数格子最容易：横着数一格就是一格，竖着数也一样。如果两点在同一条街（同样高度），距离就是横差 |x₂ − x₁|；在同一条竖线上，距离就是竖差 |y₂ − y₁|——直接减。可一旦斜着放，就得先画一个直角三角形：横差 3 格、竖差 4 格是两条直角边，斜边（真正的距离）还是问号。On the grid, counting squares is easy: one square across is one unit, and the same going up. If two points sit on the same street (same height), the distance is the gap |x₂ − x₁|; on the same vertical line, it is |y₂ − y₁| — plain subtraction. But once the points are slanted, draw a right triangle first: across 3 and up 4 are the two legs, and the diagonal — the real distance — is still the question mark.
两条直角边到手了，问号处的斜边怎么求？Two legs are ready — so how long is the hypotenuse at the question mark?
第 2 步：斜边交给勾股定理Step 2: Hand the hypotenuse to Pythagoras
直角三角形有个铁律（勾股定理）：两条直角边的平方和，正好等于斜边的平方。这里 3² + 4² = 9 + 16 = 25，开根号就是斜边 5——两点距离 5 格。把点的坐标代进去，就得到万能的距离公式：d = √((x₂ − x₁)² + (y₂ − y₁)²)。先算横差和竖差，各自平方，加起来，再开根号。Right triangles obey one iron rule (the Pythagorean theorem): leg² + leg² equals hypotenuse². Here 3² + 4² = 9 + 16 = 25, and the square root gives 5 — the distance is 5 squares. Put the coordinates in and you get the universal distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²). Take the two gaps, square each, add them, then take the square root.
这个公式，真实世界里谁在用？Who actually uses this formula in the real world?
第 3 步：无人机抄近道Step 3: The drone takes the shortcut
导航软件算「直线距离」、无人机规划航线、游戏里判断两个角色离多远——用的都是它。送快递的无人机从 A 直线飞到 B：直线距离 5 公里，比沿街绕路 7 公里省时又省电。两点之间，直线最短；要算这条直线，就回到勾股定理。Navigation apps show the "straight-line distance", delivery drones plan flight paths, games check how far two characters are — all with the same formula. The delivery drone flies straight from A to B: 5 km, versus a 7 km detour along the streets — faster and cheaper. Between two points, the straight line is shortest; to measure it, go back to Pythagoras.
🎮 你来量一量（1 分钟）🎮 Your turn: measure it (1 minute)
道理讲完了。拖动滑块摆出任务要求的两点，看水平差、竖直差和斜边距离怎么实时变化——4 个任务全过就撒花。Theory done. Move the sliders to set up the two points each task asks for, and watch the gaps and the distance update live — clear all 4 tasks to set off the confetti.
一句话记住它：两点距离 = 横差、竖差拼成的直角三角形的斜边；勾股定理一开根号就到手。Remember it in one line: the distance is the hypotenuse of the right triangle built from the two gaps — Pythagoras plus a square root, done.
横平竖直：直接减，|x₂ − x₁| 或 |y₂ − y₁|Flat or upright: just subtract — |x₂ − x₁| or |y₂ − y₁|斜着走：横差、竖差是两条直角边，距离是斜边Slanted: the two gaps are the legs, the distance is the hypotenuse距离公式：d = √((x₂ − x₁)² + (y₂ − y₁)²)Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²)
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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