👯 全等与相似：一模一样 vs 放大缩小👯 Congruence & Similarity: Identical vs Scaled
上一片里，3-4-5 的直角三角板帮机器人量出了直角。工地上还有两件“找茬”任务：有的图形一模一样，有的只是放大缩小——分清楚它们，才能照图纸正确放大施工。Last leaf, the 3-4-5 set square gave the robot a perfect right angle. Now two look-alike tasks await on site: some shapes are exact copies, others are scaled versions — telling them apart is how you enlarge a blueprint correctly.
第 1 步：全等——一模一样Step 1: Congruent — exact copies
先看打印店刚出炉的三张照片：哪两张一模一样，哪一张只是变大？Look at three fresh prints from the shop: which two are exact copies, and which one is just bigger?
同样的照片打印三份：其中两张连大小都一样，一张转了方向也没关系——平移、旋转、翻转之后能完全重合，就叫全等（≅）。全等图形对应边相等、对应角也相等，形状和大小都不变。第三张是同款照片的放大版：形状一模一样，只是尺寸按比例变了——那叫相似（∽）。The same photo printed three times: two prints match in size — even the turned one overlaps exactly after a slide, a turn or a flip. Shapes that coincide after such moves are congruent (≅): matching sides, matching angles, same shape and same size. The third print is the same photo enlarged — same shape, different size. That is similar (∽).
一模一样的好认，那“只是放大”的在生活里长什么样？Exact copies are easy to spot — where does “just scaled” show up in life?
第 2 步：相似——影子测树高Step 2: Similar — measuring a tree by its shadow
太阳光下，大树和木棍的影子各自围出一个直角三角形，两个三角形形状完全相同，只有大小按比例不同——这就是相似三角形。树高 : 棍高 = 树影 : 棍影：1 米高的棍子影长 1.5 米，6 米长的树影对应的树高就是 4 米。不用爬树，量影子就够了。In the sunlight, the tree and the stick each cast a shadow that forms a right triangle — the same shape exactly, only scaled by a ratio. Those are similar triangles. Tree height : stick height = tree shadow : stick shadow: a 1 m stick with a 1.5 m shadow, and a 6 m tree shadow, give a tree height of 4 m. No climbing needed — just measure the shadow.
相似的图形都按比例缩放，那这个“比例”还管着什么？Similar shapes all scale by a ratio — what else does that ratio control?
第 3 步：地图缩尺——相似比与面积比Step 3: Map scale — length ratio vs area ratio
地图就是一整片大地的相似缩小版：比例尺 1 : 10000 表示地图上 1 厘米 = 实际 100 米。注意面积的算法：边长放大 k 倍，面积就放大 k² 倍。地图上 1 平方厘米，实际就是 100 米 × 100 米 = 1 公顷。看边长比，也别忘了平方。A map is the whole landscape scaled down: a scale of 1 : 10000 means 1 cm on the map = 100 m in reality. Mind the areas: when lengths grow k times, areas grow k² times. One square centimetre on the map is 100 m × 100 m = 1 hectare in reality. Scale the sides — and don't forget to square for the area.
🎮 你来当辨认员（30 秒）🎮 Your turn: the shape inspector (30 seconds)
道理讲完了。看两张图形卡，判断这一对是全等还是相似——形状、大小都要看。连过 4 轮。Theory done. Look at each pair of shape cards and decide: congruent or similar? Check both shape and size. Clear 4 rounds.
一句话记住它：全等 = 形状、大小都相同（平移旋转翻转能重合，≅）；相似 = 形状相同、大小按比例（∽）。Remember it in one line: congruent = same shape and same size (slides, turns and flips still overlap, ≅); similar = same shape scaled by a ratio (∽).
全等 ≅：一模一样，只是位置或朝向不同Congruent ≅: exact copies, only moved or turned相似 ∽：同形状按比例缩放，对应边成比例、对应角相等Similar ∽: same shape scaled; sides in ratio, angles equal边长比 k → 面积比 k²：地图 1 cm² = 实际 1 公顷Length ratio k → area ratio k²: 1 cm² on a 1:10000 map = 1 hectare
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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