📚 Enlargement and Scale Factor | 放大与比例尺因子
An enlargement is a transformation that changes the size of a shape. The shape may become bigger or smaller, but its proportions and angles stay the same. Enlargements are described by a centre of enlargement and a scale factor. This topic is essential in the Cambridge KS3 mathematics curriculum and builds the foundation for similarity and scale drawings.
放大是一种改变图形大小的变换。图形的形状可能变大或变小，但比例和角度保持不变。放大由放大中心和比例尺因子来描述。这一主题是剑桥 KS3 数学课程的重要内容，为相似形和比例绘图打下基础。
1. What Is an Enlargement? | 什么是放大？
An enlargement is a geometric transformation that maps an object to an image of a different size. The image is similar to the original object – corresponding angles are equal, and all lengths are multiplied by the same scale factor. Unlike reflection, rotation or translation, enlargement changes size.
放大是一种几何变换，它将原图形映射为不同大小的影像。影像与原物相似——对应角相等，所有长度都乘以相同的比例因子。与反射、旋转或平移不同，放大改变了尺寸。
To describe an enlargement fully, you must state the centre of enlargement and the scale factor. For example, an enlargement with scale factor 3 makes all sides three times longer.
要完整描述一次放大，必须注明放大中心和比例因子。例如，比例因子为 3 的放大会使所有边长变为原来的三倍。
2. Centre of Enlargement | 放大中心
The centre of enlargement is the fixed point from which rays are drawn through the vertices of the object. The image points lie on these rays, and their distances from the centre are multiplied by the scale factor. If the centre is at a vertex of the shape, that vertex does not move.
放大中心是一个固定点，从该点出发通过原图形顶点画射线。影像点位于这些射线上，它们到中心的距离乘以比例因子。如果中心在图形的某个顶点，则该顶点不动。
You can find the image of a point P by drawing a line from the centre C through P, then measuring the distance CP and multiplying by the scale factor k. The image point P′ lies on the same line such that CP′ = k × CP.
找到点 P 的影像，可以从中心 C 画一条经过 P 的直线，测量距离 CP 并乘以比例因子 k。影像点 P′ 在同一直线上，且满足 CP′ = k × CP。
3. Scale Factor – Magnification and Reduction | 比例因子 – 放大与缩小
The scale factor (k) determines how much the object is enlarged. If k > 1, the image is larger than the object – this is a true enlargement or magnification. If 0 < k < 1, the image is smaller than the object – we call this a reduction. A scale factor of 1 leaves the shape unchanged.
比例因子 (k) 决定原图形放大的程度。如果 k > 1，影像比原物大，这是真正的放大。如果 0 < k < 1，影像比原物小，我们称之为缩小。比例因子为 1 时形状不变。
For example, when k = 2, the image is twice as wide and twice as tall. When k = ½, the image is half the size of the original. The shape remains identical in its proportions, only the size changes.
例如，当 k = 2 时，影像的宽和高都扩大为两倍。当 k = ½ 时，影像是原图形大小的一半。图形的比例保持不变，只有尺寸发生变化。
4. Enlargement Using Coordinate Grids | 在坐标网格上进行放大
When the centre of enlargement is at the origin (0,0), enlarging a point (x, y) with scale factor k gives the image point (kx, ky). This is the simplest case and is very common in Cambridge KS3 exercises.
当放大中心在原点 (0,0) 时，用比例因子 k 放大点 (x, y) 可得到影像点 (kx, ky)。这是最简单的情况，在剑桥 KS3 练习中十分常见。
If the centre is not at the origin, you can use vector method: find the vector from the centre to the point, multiply by k, then add back to the centre coordinates. Example: centre (2,1), point (5,3). Vector = (3,2). Multiply by k=3 → (9,6). Image = (2+9, 1+6) = (11,7).
如果中心不在原点，可以使用向量法：求出从中心到点的向量，乘以 k，再加回中心坐标。例：中心 (2,1)，点 (5,3)。向量 = (3,2)。乘以 k=3 → (9,6)。影像 = (2+9, 1+6) = (11,7)。
5. Drawing an Enlargement Step by Step | 逐步绘制放大图形
Follow these steps to construct an enlargement: (1) Mark the centre of enlargement C. (2) Draw straight rays from C through each vertex of the object. (3) Measure the distance from C to a vertex, multiply by k, and mark the image point on the same ray. (4) Repeat for all vertices. (5) Join the image vertices to form the enlarged shape.
按照以下步骤构建放大图形：(1) 标出放大中心 C。(2) 从 C 出发通过原图形每个顶点画射线。(3) 测量 C 到顶点的距离，乘以 k，在同一条射线上标出影像点。(4) 对所有顶点重复。(5) 连接影像顶点，形成放大后的图形。
When k is a fraction like ½, the image points are closer to the centre than the original vertices. Always label your object and image clearly, and write down the scale factor used.
当 k 为 ½ 这样的分数时，影像点比原来的顶点更靠近中心。始终清晰地标注原物和影像，并写明所使用的比例因子。
6. Finding the Centre of Enlargement | 找出放大中心
If you are given an object and its image but not the centre of enlargement, you can find it by drawing lines through corresponding vertices. The lines should all intersect at a single point – that is the centre of enlargement. This works because all rays from the centre pass through corresponding points.
如果已知原图形及其影像，但尚未给出放大中心，可以通过连接对应顶点画直线来找到中心。所有这些直线应交于同一点——那就是放大中心。这是因为从中心发出的所有射线都经过对应点。
For example, draw a line through A and its image A′, and another through B and B′. Their intersection is the centre of enlargement. If the lines do not meet at a single point, the transformation is not an enlargement.
例如，画一条经过 A 及其影像 A′ 的直线，再画一条经过 B 和 B′ 的直线。这两条线的交点就是放大中心。如果多条直线没有交于同一点，则该变换不是放大。
7. Calculating the Scale Factor from a Diagram | 从图形中计算比例因子
To find the scale factor k when the centre is known, measure a length on the image and divide by the corresponding length on the object: k = image length ÷ object length. Always use the same pair of corresponding sides.
当中心已知时，要找出比例因子 k，只需测量影像上某一段长度，除以原图形上对应段的长度：k = 影像长度 ÷ 原图形长度。务必使用同一对对应边。
In coordinate geometry, if the centre is the origin, simply compare the coordinates of a vertex and its image: k = x-coordinate of image / x-coordinate of object (provided the x-coordinate is non-zero). You should get the same value for both x and y.
在坐标几何中，如果中心是原点，只需比较某个顶点的坐标及其影像的坐标：k = 影像的 x 坐标 / 原物的 x 坐标（假设 x 坐标非零）。x 和 y 方向应得出相同的值。
8. Enlargement with Fractional Scale Factors (Reduction) | 分数比例因子的放大（缩小）
When the scale factor is between 0 and 1, the image is smaller than the object. For instance, k = ½ halves every length. The image is still similar, and the centre of enlargement still works in the same way. Reduction is just a special case of enlargement.
当比例因子在 0 和 1 之间时，影像比原物小。例如，k = ½ 使每条边减半。影像仍然相似，放大中心的作用方式不变。缩小只是放大的一种特殊情况。
To draw a reduction with centre C and k = ½, measure the distance from C to a vertex, halve it, and mark the image point. The shape will be squeezed towards the centre but will maintain its angles.
要绘制以 C 为放大中心、k = ½ 的缩小图形，测量从 C 到顶点的距离，取一半，标出影像点。图形会向中心收缩，但保持所有角度。
9. Enlargement with Negative Scale Factors (Extension) | 负比例因子的放大（拓展）
For Cambridge KS3 advanced learners, it’s useful to know that scale factors can also be negative. An enlargement with a negative scale factor, e.g. k = -2, produces an image on the opposite side of the centre. The size is multiplied by |k|, but the image is inverted.
对于剑桥 KS3 学有余力的学生，了解比例因子也可以为负数是很有用的。使用负比例因子的放大，如 k = -2，会在中心的另一侧产生影像。尺寸乘以 |k|，但影像方向相反。
If k = -1, the image is the same size but rotated 180° about the centre; this is equivalent to a half-turn. Standard KS3 may not require this, but it builds a deeper understanding.
若 k = -1，影像大小相同，但绕中心旋转 180°；这等效于半周旋转。常规 KS3 可能不要求掌握，但它有助于深入理解。
10. Real-Life Applications of Enlargement | 放大在现实生活中的应用
Enlargement principles are used in map scales, architectural blueprints, and digital zooming. A scale model of a building is a classic example of reduction, where the scale factor is a fraction like 1:50. Understanding enlargement helps in reading and creating scale drawings.
放大的原理应用于地图比例尺、建筑蓝图和数字变焦。建筑的比例模型是缩小的典型例子，比例因子如 1:50 这样的分数。理解放大有助于阅读和制作比例图。
Another application is in photocopiers: enlarging a document by 200% means k = 2, while reducing to 50% means k = 0.5. The centre of enlargement corresponds to the point that stays in place during scanning.
另一个应用是复印机：将文档放大 200% 即 k = 2，缩小到 50% 即 k = 0.5。放大中心对应于扫描过程中保持不动的点。
11. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Forgetting to multiply all distances: Some students only enlarge the horizontal or vertical distances from the centre. Always move along the ray and multiply the exact distance. Wrong centre: Using a different point as the centre will give an incorrect image. Double-check the given centre. Confusing length and scale factor: The scale factor is the multiplier, not the image length itself.
忘记将所有距离相乘：有些学生只将中心出发的水平或竖直距离放大。应始终沿射线移动，并乘以实际的距离。错误的中心：以别的点作为中心会得到不正确的影像。请仔细检查给定的中心。混淆长度与比例因子：比例因子是乘数，而不是影像长度本身。
Practice with different centres on grid paper, and measure accurately. Always compare corresponding side lengths to verify your scale factor.
在方格纸上用不同中心多加练习，并精确测量。始终比较对应边长以验证你的比例因子。
12. Revision Checklist for Enlargements | 放大知识复习清单
- I can define enlargement and identify the centre and scale factor. 我能定义放大并识别中心和比例因子。
- I can enlarge a shape given a centre and positive scale factor >1. 我能根据给定的中心和大于 1 的正比例因放大图形。
- I can perform a reduction using a fractional scale factor between 0 and 1. 我能用 0 和 1 之间的分数比例因子进行缩小。
- I can find the centre of enlargement from an object and its image. 我能根据原图形及其影像找出放大中心。
- I can calculate the scale factor from a diagram or coordinates. 我能从图形或坐标中计算比例因子。
- I can solve problems involving negative scale factors (extension). 我能解答涉及负比例因子的问题（拓展）。
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