变换与对称

考什么 · What it tests

需要先会 · Prerequisites

• 坐标系与点的表示 / The coordinate plane and points
• 角度与旋转方向 / Angles and direction of rotation
• 全等（变换前后图形全等） / Congruence (shapes stay congruent under these moves)

常见套路 · Common moves

1. 变换保持全等：平移、旋转、反射都不改变形状和大小，对应边角相等——先认定「变换前后全等」再往下推。

   These transformations preserve congruence: translation, rotation, and reflection keep shape and size, so corresponding sides and angles are equal — lock in "congruent before and after" first.
2. 对称当镜子省一半：图形关于某条线对称时，只算一半再翻倍；求最短路径时把点「照镜子」反射过去拉成直线。

   Use symmetry as a mirror to save half: when a figure is symmetric about a line, compute one half and double it; for a shortest path, reflect a point across the mirror to straighten the route into a line.
3. 旋转抓「旋转中心 + 角度」：绕定点转固定角度，到旋转中心的距离不变，常把分散的图形转到一起拼成规则形。

   For rotation, fix the center and angle: turning about a point through a set angle keeps distances to the center, and it often rotates scattered pieces together into a regular shape.

易错点 · Common pitfalls

• 反射方向搞反或对称轴找错：翻折后的点和原点到对称轴距离相等、连线垂直于轴，别只是平移过去。

  Reflecting the wrong way or picking the wrong axis: the image and original are equidistant from the axis with their connector perpendicular to it — it's not just a slide.
• 旋转方向（顺/逆时针）和角度记错：$90°$ 和 $180°$ 结果差很多，先定好中心和方向。

  Getting the rotation's direction (clockwise/counter) or angle wrong: $90°$ and $180°$ land very differently, so fix the center and direction first.