📚 Transformations - Reflections, Rotations and Translations | 变换：反射、旋转和平移
Welcome to your guide on transformations, a key topic in the Cambridge KS3 Mathematics curriculum. On p210 of your textbook, you will encounter how shapes move and change position on a grid without altering their size or shape. This article covers reflections, rotations, translations, and combinations of these movements, giving you the tools to describe and carry out each transformation accurately.
欢迎阅读本篇变换指南——这是剑桥 KS3 数学课程中的一个核心主题。在教材第 210 页，你将学习图形如何在网格上移动和改变位置，而不会改变其大小或形状。本文涵盖反射、旋转和平移，以及这些运动的组合，为你提供准确描述和执行每种变换的方法。
1. Understanding Rigid Transformations | 理解刚体变换
In geometry, a rigid transformation (also called an isometry) moves a shape so that all corresponding lengths and angles remain the same. The shape's size and shape never change — only its position and orientation. The three main rigid transformations at KS3 are translation, reflection, and rotation.
在几何中，刚体变换（也称为等距变换）移动图形时，所有对应的长度和角度都保持不变。图形的大小和形状永远不会改变——只有位置和方向发生变化。KS3阶段涉及的三个主要刚体变换是平移、反射和旋转。
Because rigid transformations preserve length and angle, the image you get after a transformation is congruent to the original object. Congruence means you could cut out the original and place it exactly on top of the image, even if you needed to flip or turn it.
因为刚体变换保持长度和角度，变换后得到的像与原图形全等。全等意味着你可以把原图形剪下来，完全覆盖在像上，哪怕需要翻转或旋转它。
2. Mastering Translation | 掌握平移
nd the bottom number tells you how many units vertically (positive = up, negative = down).
平移将图形上的每一点，沿着指定方向移动相同距离。你用一个列向量来描述平移，例如 (3 −²)，其中上面的数字表示水平移动的单位数（正数=向右，负数=向左），下面的数字表示垂直移动的单位数（正数=向上，负数=向下）。
On a coordinate grid, if you apply translation vector (a b) to a point (x, y), the image point is (x+a, y+b). KS3 questions often ask you to translate a shape and label the new vertices, or to work out the vector from the object to the image.
在坐标网格中，如果将平移向量 (a b) 应用于点 (x, y)，其像点坐标为 (x+a, y+b)。KS3 题目常常要求你平移某个图形并标记新的顶点，或者根据原图形与像求出平移向量。
Common mistakes include swapping the order of coordinates in the vector or forgetting to add the vector to every vertex. Always keep the shape exactly the same; if you accidentally change side lengths, you have stretched the shape, which is not a translation.
常见的错误包括交换向量中两个分量的顺序，或者忘记将向量加到每一个顶点上。始终要保持图形完全一致；如果不小心改变了边长，就相当于拉伸了图形，那就不再是平移。
3. Exploring Reflection | 探索反射
A reflection produces the mirror image of a shape across a given line, called the mirror line. Under reflection, each point and its image lie on opposite sides of the mirror line, at equal perpendicular distances. The shape flips, but its size stays unchanged.
反射是图形关于某条给定直线（称为镜面线）所作出的镜像。在反射下，每一个点与其像点分居镜面线两侧，且到镜面线的垂直距离相等。图形会发生翻转，但大小不变。
u will learn to measure distances with a ruler on plain paper and count squares on a grid.
KS3 阶段典型的镜面线包括 x 轴、 y 轴、直线 y = x 和 y = −x，以及任意水平或竖直线，如 x = 2 或 y = −1。你将学习在方格纸上用直尺测量距离，并在网格上数格子。
x). Practice drawing the mirror line and checking that corresponding points are symmetric.
当点 (x, y) 关于 y 轴反射时，像变为 (−x, y)；关于 x 轴反射时，像变为 (x, −y)。关于直线 y = x 反射时，坐标互换：(y, x)。练习画出镜面线，并检查对应点是否对称。
4. Carrying Out Rotations | 执行旋转
A rotation turns a shape around a fixed point, called the centre of rotation. You must specify three things to describe a rotation fully: the angle of rotation (e.g. 90°, 180°), the direction (clockwise or anticlockwise), and the centre.
旋转使图形绕着一个固定点转动，这个点称为旋转中心。要完整描述一个旋转，必须明确三要素：旋转角度（如 90°、 180°）、方向（顺时针或逆时针）以及旋转中心。
At KS3, centres are often the origin (0,0) or other points on a grid. 90° rotations move a shape a quarter turn, 180° a half turn. Tracing paper is a brilliant tool — you can trace the shape, pin the centre with a pencil, and turn the paper to see where the image lands.
在 KS3 阶段，旋转中心通常是原点 (0,0) 或网格上的其他点。90° 旋转相当于四分之一圈，180° 相当于半圈。描图纸是非常管用的工具——你可以描出图形，用铅笔尖固定旋转中心，然后转动描图纸，就可看到像应落在何处。
(y, −x). Notice how coordinates swap and signs change depending on the angle and direction.
如果将点 (x, y) 绕原点旋转 180°，像为 (−x, −y)。绕原点逆时针旋转 90° 时，(x, y) → (−y, x)；顺时针旋转 90° 时，(x, y) → (y, −x)。注意坐标如何互换以及正负号怎样根据角度和方向变化。
5. Distinguishing Between Transformations | 区分不同变换
It is essential to recognize which transformation you are dealing with. A translation slides the shape without turning or flipping. A reflection creates a mirror image, often changing the orientation. A rotation turns the shape and can move it into a new position while keeping the same orientation relative to the centre.
能够识别所处理的变换类型至关重要。平移是滑动图形，不转动也不翻转。反射产生镜像，通常会改变图形朝向。旋转会转动图形，并相对于旋转中心把图形移到新位置，但保持形状的固有朝向不变。
KS3 exams often give two congruent shapes on a grid and ask you to describe the single transformation that maps the first onto the second. Look for evidence: if the shape faces the same way but has shifted, it is a translation. If the shape is flipped and the distances to a given line are equal, go for reflection. If the shape has turned, it is a rotation; find the centre by joining corresponding points and looking for angles.
KS3 考试中经常给出网格上的两个全等图形，要求描述将一个图形映射到另一个图形的单一变换。寻找证据：如果图形朝向相同但位置移动了，那是平移。如果图形翻转了，而且到某条直线的距离相等，那就选反射。如果图形转动了，那就是旋转；通过连接对应点并寻找角度可以找到旋转中心。
6. Working with Multiple Transformations | 进行复合变换
Sometimes you are asked to carry out two transformations in sequence, for example first a reflection and then a translation. Apply the first transformation to get an intermediate image, then apply the second transformation to that image. Label each step clearly so you do not lose track.
有时候题目要求依次执行两个变换，例如先反射再平移。先执行第一个变换得到中间像，再对这个中间像执行第二个变换。每一步都要清晰地标记，以免混淆。
When describing a combination, you may use language like ‘reflect in the line y = 1, then translate by vector (−3 2)’. The order matters — ecting generally gives a different final position. Try both orders on a sketch to see why.
描述复合变换时，可能会用到这样的表述：“先关于直线 y = 1 反射，再以向量 (−3 2) 平移”。顺序很重要——先平移再反射，通常会得到不同的最终位置。试着在草图上按两种顺序各做一遍，就能明白为什么。
7. Describing Rotations Accurately | 准确地描述旋转
To describe a rotation, you must give the angle, direction, and centre. If the question shows a shape rotated by 90° clockwise about (2,1), writing ‘rotation 90°’ loses marks. Always check if the centre is the origin or a different point on the grid.
要描述旋转，必须给出角度、方向和旋转中心。如果题目展示了一个图形绕点 (2,1) 顺时针旋转 90°，只写“旋转 90°”是会丢分的。一定要检查旋转中心是原点还是网格上的其他点。
You can find the centre of rotation by connecting corresponding vertices from object and image with straight line segments, then drawing the perpendicular bisectors. The intersection of those bisectors is the centre. At KS3, you might use tracing paper to test your guess.
你可以通过连接原图形和像中对应的顶点，画出线段，再作这些线段的中垂线，来找到旋转中心。中垂线的交点就是旋转中心。在 KS3 阶段，你也可以用描图纸来检验自己的猜测。
8. Understanding Invariant Points | 理解不动点
An invariant point is a point that does not move under a transformation. In a reflection, every point on the mirror line stays fixed. In a rotation, only the centre of rotation is invariant (unless the angle is a multiple of 360°). In a translation with a non-zero vector, no point is invariant.
不动点是指在变换下位置保持不变的点。在反射中，镜面线上的每一个点都固定不动。在旋转中，只有旋转中心是不动点（除非旋转角度是 360° 的倍数）。当平移向量非零时，平移没有不动点。
KS3 questions may ask, ‘Which points remain unchanged after reflection in the y-axis?’ Points on the y-axis have x-coordinate 0, so they map to themselves. Recognising invariant points deepens your understanding of each transformation.
KS3 题目可能会问：“关于 y 轴反射后，哪些点保持不变？”此时 y 轴上的点 x 坐标为 0，会映射到自身。识别不动点可以加深你对每种变换的理解。
9. Transformations on a Coordinate Grid | 坐标网格上的变换
Working on a grid helps you count squares and verify coordinates. Always write the coordinates of the object’s vertices before and after the transformation in a logical order. Label the image vertices with prime notation, e.g. A′, B′, C′.
在网格上操作有助于你数格子并验证坐标。一定要按有条理的顺序写出原图形各顶点变换前后的坐标。将像的顶点用带撇的记号标注，如 A′、 B′、 C′。
For translations, simply add the vector to each vertex. For reflections, count how far each vertex is from the mirror line and plot the image the same distance out on the other side. For rotations, use the centre and angle rules or tracing paper to find new coordinates.
平移时，只需把向量加到每一个顶点上。反射时，数一下每个顶点到镜面线的距离，然后在另一侧等距离处描出像点。旋转时，使用旋转中心与角度规则，或借助描图纸找出新坐标。
10. Symmetry, Patterns and Transformations | 对称性、图案与变换
Reflection and rotation connect closely to line symmetry and rotational symmetry. A shape with reflection symmetry has one or more mirror lines where one half is the exact reflection of the other. A shape with rotational symmetry can be rotated less than 360° about its centre and still fit exactly onto itself.
反射和旋转与轴对称和旋转对称密切相关。具有反射对称性的图形有一条或多条镜面线，使得一半恰为另一半的反射。具有旋转对称性的图形可以围绕其中心旋转小于 360°，而仍与自身完全重合。
Transformations are used to generate patterns and tessellations. For instance, repeating a tile by translating it produces a simple block pattern, while adding reflections or rotations creates more intricate designs. Understanding this link helps in both geometry and art.
变换可用于生成图案和密铺。例如，通过平移重复一个基本瓦片可以产生简单的块状图案，而添加反射或旋转可以创造出更为复杂的纹样。理解这种联系对几何和艺术都有帮助。
11. Common Pitfalls and How to Avoid Them | 常见陷阱与避免方法
ncorrect mirror line. A reflection in x = 1 is not the same as x = −1, so check carefully.
错误 1：混淆顺时针与逆时针。务必重读题目并画出箭头确认方向。错误 2：镜面线错误。关于 x = 1 的反射与 x = −1 不同，要仔细检查。
Mistake 3: forgetting to list the centre in a rotation description. A rotation about (0,0) is different from one about (2,3), so the centre is vital. Mistake 4: using the wrong column vector. Some students write the vertical shift as the top number; remember the horizontal shift always comes first.
错误 3：描述旋转时忘记列出旋转中心。关于原点的旋转与关于 (2,3) 的旋转不同，因此中心必不可少。错误 4：列向量写反。有些学生把垂直移动作为向量的第一个分量；请记住，水平移动始终写在上面。
12. Summary and Revision Tips | 总结与复习技巧
Transformations are everywhere in KS3 maths. Translating, reflecting and rotating shapes builds spatial reasoning and prepares you for more advanced topics like vectors, similarity, and circle theorems in later years. Use tracing paper and grid practice to build confidence.
变换在 KS3 数学中无处不在。平移、反射和旋转图形可以培养空间推理能力，并为今后学习向量、相似性和圆定理等更高阶内容打下基础。利用描图纸和网格练习来建立信心。
When revising, create flashcards for each transformation with its key properties and coordinate rules. Test yourself by drawing a triangle, applying a transformation, and then describing it in full. This active recall cements the concepts for your exams.
复习时，可以为每种变换制作卡片，写上关键性质和坐标规则。给自己出题：画一个三角形，进行某种变换，然后完整地描述出来。这种主动回忆的方法能巩固概念，帮助你在考试中发挥出色。
Published by TutorHao | KS3 Cambridge Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图