📚 Position and Movement: Translating Shapes | 位置与移动：平移
In this revision guide, we explore translations on a coordinate grid. You will learn how to describe a translation, use a translation vector, and move points and shapes accurately. This topic is central to Cambridge KS3 Mathematics and builds a foundation for later transformation geometry.
在本复习指南中，我们将学习坐标网格中的平移。你将学会如何描述平移、使用平移向量，并准确地移动点和图形。该主题是剑桥 KS3 数学的核心内容，为后续的变换几何打下基础。
1. Coordinates and the Four Quadrants | 坐标与四个象限
A coordinate grid is formed by a horizontal x-axis and a vertical y-axis. The point where they meet is the origin, written as (0, 0). Positions are given as ordered pairs (x, y). The x-coordinate tells you how far to move left or right; the y-coordinate tells you how far to move up or down.
坐标网格由水平的 x 轴和竖直的 y 轴组成。两轴相交的点称为原点，记作 (0, 0)。位置用有序数对 (x, y) 表示。x 坐标表示左右移动的距离，y 坐标表示上下移动的距离。
In Key Stage 3, you work in all four quadrants. In the first quadrant both coordinates are positive. In the second quadrant x is negative and y is positive. In the third quadrant both are negative. In the fourth quadrant x is positive and y is negative.
在 KS3 阶段，你需要在四个象限中作图。第一象限两个坐标都为正；第二象限 x 为负、y 为正；第三象限两个都为负；第四象限 x 为正、y 为负。
2. What is a Translation? | 什么是平移
A translation is a transformation that moves every point of a shape the same distance in the same direction. It does not rotate, reflect, or resize the shape. The shape after a translation is called the image.
平移是一种变换，它使图形上的每个点沿相同方向移动相同的距离。平移不旋转、不反射、也不改变大小。平移后的图形称为像。
Because a translation only changes position, the original shape and its image are congruent. They have exactly the same side lengths, angles, and area.
由于平移只改变位置，原图形与其像是全等的。它们具有完全相同的边长、角度和面积。
3. Describing a Translation with a Vector | 用向量描述平移
A translation is described by a vector. In school work, we usually write a vector as a column vector or in the form (a, b), where a is the horizontal movement and b is the vertical movement.
平移用向量来描述。在中学作业中，通常把向量写成列向量或 (a, b) 的形式，其中 a 表示水平移动，b 表示竖直移动。
If a is positive, the shape moves right. If a is negative, it moves left. If b is positive, the shape moves up. If b is negative, it moves down.
如果 a 为正，图形向右移动；如果 a 为负，图形向左移动。如果 b 为正，图形向上移动；如果 b 为负，图形向下移动。
For example, the vector (4, -3) means "move 4 units right and 3 units down".
例如，向量 (4, -3) 表示 "向右移动 4 个单位，向下移动 3 个单位"。
4. Translating a Point | 平移一个点
To translate a point (x, y) by a vector (a, b), add a to the x-coordinate and b to the y-coordinate.
要用向量 (a, b) 平移一个点 (x, y)，需要把 a 加到 x 坐标上，把 b 加到 y 坐标上。
(x, y) → (x + a, y + b)
For example, translate the point (2, 5) by the vector (3, -1).
例如，用向量 (3, -1) 平移点 (2, 5)。
Add 3 to 2 to get 5 for the new x-coordinate. Add -1 to 5 to get 4 for the new y-coordinate. So the image is (5, 4).
把 2 加 3 得到新的 x 坐标 5；把 5 加 -1 得到新的 y 坐标 4。所以像点为 (5, 4)。
5. Translating a Shape | 平移一个图形
To translate a polygon, apply the same vector to each vertex. Then join the image vertices in the same order. This guarantees the image is congruent to the original shape.
要平移一个多边形，需要对每个顶点应用相同的向量。然后按相同顺序连接像顶点。这可以保证像与原图形全等。
For example, translate triangle ABC with vertices A(1, 2), B(3, 5), C(4, 1) by the vector (-2, 3).
例如，用向量 (-2, 3) 平移三角形 ABC，其顶点为 A(1, 2)、B(3, 5)、C(4, 1)。
A moves to (-1, 5), B moves to (1, 8), and C moves to (2, 4). Plot these image points and connect them.
A 移动到 (-1, 5)，B 移动到 (1, 8)，C 移动到 (2, 4)。标出这些像点并连接它们。
| Original point | Image point |
| A(1, 2) | A'(-1, 5) |
| B(3, 5) | B'(1, 8) |
| C(4, 1) | C'(2, 4) |
6. Writing a Vector from a Translation | 根据平移写出向量
If you are given the original point and its image, you can find the translation vector. Subtract the original coordinates from the image coordinates.
如果给出原点和它的像点，你可以求出平移向量。用像点坐标减去原点坐标即可。
a = x-image − x-original, b = y-image − y-original
For example, point P(3, -1) maps to P'(7, 2). Find the translation vector.
例如，点 P(3, -1) 映射到 P'(7, 2)。求平移向量。
a = 7 − 3 = 4, b = 2 − (−1) = 3. The vector is (4, 3).
a = 7 − 3 = 4，b = 2 − (−1) = 3。向量为 (4, 3)。
7. Congruence and Invariance | 全等性与不变性
Under a translation, side lengths and angle sizes remain unchanged. Only the position changes. Therefore, translation is an isometry, also called a congruence transformation.
在平移下，边长和角度大小保持不变，只有位置改变。因此，平移是一种等距变换，也称为全等变换。
This means you can use a translation to prove two shapes are congruent. If one shape is an exact translated copy of another, they are congruent.
这意味着你可以用平移来证明两个图形全等。如果一个图形是另一个图形的精确平移副本，那么它们全等。
8. Combining Translations | 组合平移
If a shape is translated first by vector (a, b) and then by vector (c, d), the combined translation is (a + c, b + d). You add the horizontal parts and the vertical parts separately.
如果图形先按向量 (a, b) 平移，再按向量 (c, d) 平移，那么组合平移为 (a + c, b + d)。水平部分相加，竖直部分相加。
(a, b) + (c, d) = (a + c, b + d)
For example, (3, -2) followed by (-1, 5) gives (2, 3). The order does not matter because vector addition is commutative.
例如，(3, -2) 后再 (-1, 5) 得到 (2, 3)。顺序无关紧要，因为向量加法满足交换律。
9. Common Mistakes | 常见错误
A common mistake is adding the wrong sign. Remember that negative a means left and negative b means down.
一个常见错误是加错符号。记住，a 为负表示向左，b 为负表示向下。
Another mistake is moving only one vertex of a shape. Every vertex must be translated by the same vector, otherwise the shape will be distorted.
另一个错误是只移动图形的一个顶点。每个顶点都必须按同一向量平移，否则图形会变形。
Students sometimes confuse the x- and y-coordinates in a column vector. The first number is always the horizontal movement, and the second number is the vertical movement.
学生有时会混淆列向量中的 x 和 y 坐标。第一个数总是水平移动，第二个数总是竖直移动。 Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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