📚 Describing Enlargements | 描述放大
In geometry, an enlargement changes the size of a shape but not its proportions. When you enlarge a shape on a grid, every side length is multiplied by the same scale factor, and all the lines radiate from a fixed point called the centre of enlargement. Understanding how to describe an enlargement fully, including its scale factor and centre, is a core skill in the Cambridge KS3 mathematics curriculum. It connects ratio, proportion and coordinate geometry, and prepares you for more advanced transformations such as similar shapes and scale drawings.
在几何中，放大变换会改变图形的大小，但不会改变其比例。当你在网格上放大一个图形时，每条边的长度都会乘以相同的比例因子，而且所有的对应点连线都从一个固定的点发出，这个点叫做放大中心。完整描述一个放大，包括它的比例因子和中心，是剑桥初中数学课程的核心技能。这一内容把比、比例和坐标几何联系起来，为后续学习相似形和比例尺图打下基础。
1. What Is an Enlargement? | 什么是放大？
An enlargement is a transformation that changes the size of a shape. Unlike reflections, rotations or translations, an enlargement does not preserve side lengths – it preserves the shape’s angles and the ratios between corresponding side lengths. This means the image is mathematically similar to the original object. When you enlarge a shape, you need a scale factor (which tells you how many times bigger or smaller the image is) and a centre of enlargement (the fixed point from which the enlargement is projected).
放大是一种改变图形大小的变换。与反射、旋转或平移不同，放大不保持边的长度——它保持图形的角以及对应边长之间的比率不变。这意味着像与原物体在数学上是相似的。进行放大时，你需要一个比例因子（告诉你像是原物体的多少倍）和一个放大中心（投影放大时固定不变的点）。
2. Scale Factor | 比例因子
The scale factor, usually denoted by the letter k, is the multiplier that describes how much the shape grows or shrinks. If k > 1, the image is larger than the object. If 0 < k < 1, the image is smaller – this is sometimes called a reduction, but it is still an enlargement with a fractional scale factor. The length of any side on the image equals k times the length of the corresponding side on the object. Area, however, scales by a factor of k².
比例因子通常用字母 k 表示，是描述图形放大或缩小的倍数。如果 k > 1，像比原物大。如果 0 < k < 1，像比原物小——这有时被称为缩小，但在数学上仍然属于放大，只是使用了分数比例因子。像上任意一条边的长度等于原物对应边长乘以 k。然而，面积会按 k² 的比例变化。
3. 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 to locate the vertices of the image. All rays from the centre pass through corresponding points on the object and the image. The distance from the centre to an image vertex is k times the distance from the centre to the corresponding object vertex. The centre can be anywhere – on the shape, outside it, or even at a vertex. Its coordinates are always part of a complete description.
放大中心是一个固定点，从该点出发作射线通过原物的各个顶点，从而确定像的顶点。从中心出发的所有射线都会通过原物和像上的对应点。中心到像顶点的距离等于中心到对应原物顶点距离的 k 倍。中心可以位于任何位置——图形上、图形外，甚至一个顶点处。它的坐标始终是完整描述的一部分。
4. Positive Integer Scale Factor Enlargements | 正整数比例因子的放大
When the scale factor is a positive integer such as 2, 3 or 4, the image appears further away from the centre than the object, and all side lengths are multiplied by that integer. For example, with centre (0,0) and scale factor 2, a triangle with vertices at (1,1), (3,1) and (1,4) will map to (2,2), (6,2) and (2,8). Notice how each coordinate is simply multiplied by 2 when the centre is the origin.
当比例因子为正整数如 2、3 或 4 时，像比原物离中心更远，且所有边长都乘以该整数。例如，以 (0,0) 为中心、比例因子为 2，顶点为 (1,1)、(3,1)、(1,4) 的三角形将变为 (2,2)、(6,2) 和 (2,8)。请注意，当中心为原点时，每个坐标直接乘以比例因子即可。
5. Fractional Scale Factor Enlargements | 分数比例因子的放大
A scale factor between 0 and 1, such as ½, produces an image that is smaller than the object and lies between the centre and the object. For instance, with centre (0,0) and scale factor ½, the point (4,6) would map to (2,3). Fractional scale factors are especially useful for creating scale models and are tested regularly. Always check: if the image is smaller, the scale factor must be a proper fraction.
比例因子介于 0 和 1 之间，例如 ½，产生的像比原物小，且位于中心和原物之间。例如，以 (0,0) 为中心、比例因子为 ½，点 (4,6) 将变为 (2,3)。分数比例因子在制作比例模型时特别有用，也是常考内容。请始终记住：如果像比原物小，比例因子必定是真分数。
6. Describing an Enlargement Fully | 完整描述一个放大
In an exam, you might be asked: ‘Fully describe the single transformation that maps shape A onto shape B.’ The correct answer must include two parts: the phrase ‘enlargement’, the scale factor, and the coordinates of the centre. For example: ‘An enlargement with scale factor 3 and centre (1, -2).’ Miss one of these, and the description is incomplete, which will cost marks.
在考试中，你可能被要求：“完整描述将图形 A 映射为图形 B 的单一变换。”正确的答案必须包含两部分：“放大”一词、比例因子以及中心坐标。例如：“以 (1, -2) 为中心、比例因子为 3 的放大。” 遗漏任何一个都会导致描述不完整，从而丢分。
7. Finding the Centre of Enlargement | 寻找放大中心
To find the centre, draw rays (straight lines) through pairs of corresponding vertices from the object and the image. The point where the rays intersect is the centre of enlargement. If the image and object are aligned, it may be easy to spot; if not, use a ruler. In coordinate geometry, you can also solve algebraically. For a point A(x₁,y₁) mapping to A'(x₂,y₂), the centre (a,b) satisfies x₂ = a + k(x₁ – a) and y₂ = b + k(y₁ – b). Rearranging gives a and b.
要找到放大中心，可以通过原物和像的各对对应顶点画射线（直线）。射线的交点就是放大中心。如果像和原物整齐对应，可能很容易看出；否则需使用直尺。在坐标几何中，你也可以用代数方法求解。对于点 A(x₁,y₁) 映射为 A'(x₂,y₂)，中心 (a,b) 满足 x₂ = a + k(x₁ – a) 和 y₂ = b + k(y₁ – b)。重新整理可求得 a 和 b。
8. Drawing an Enlargement: Ray Method | 绘制放大：射线法
Given a shape, a centre C and a scale factor k, you can draw the enlargement accurately using the ray method. Step 1: draw rays from C through each vertex of the object. Step 2: measure the distance from C to an object vertex, multiply by k, and mark the image vertex on the same ray. Step 3: repeat for all vertices. Step 4: join the image vertices in the correct order to form the enlarged shape. This method works for any type of polygon and for both integer and fractional scale factors.
给定一个图形、一个中心 C 和一个比例因子 k，你可以使用射线法精确绘制放大图。步骤一：从 C 出发通过原物的每个顶点画射线。步骤二：测量 C 到原物顶点的距离，乘以 k，在同一条射线上标出像的顶点。步骤三：对所有顶点重复此操作。步骤四：按正确顺序连接像的顶点，形成放大后的图形。这个方法适用于任何多边形，也适用于整数和分数比例因子。
9. Enlargements on Coordinate Grids | 坐标网格上的放大
When the centre of enlargement is the origin (0,0), finding image coordinates is straightforward: just multiply each coordinate of the object by the scale factor. If the centre is not the origin, you can still use vector methods: the vector from the centre to an object point is multiplied by k, then added to the centre. For example, centre (2,1), scale factor 3, point (4,3): vector from centre is (2,2); multiplied by 3 gives (6,6); add to centre gives (8,7). Drawing on squared paper makes this visual.
当放大中心为原点 (0,0) 时，求像的坐标非常简单：只需将原物的每个坐标乘以比例因子。如果中心不是原点，你仍然可以使用向量方法：从中心到原物点的向量乘以 k，再加上中心坐标。例如，中心 (2,1)、比例因子 3、点 (4,3)：中心到点的向量是 (2,2)；乘以 3 得 (6,6)；加上中心得 (8,7)。在方格纸上作图会使这个过程更加直观。
10. Common Mistakes to Avoid | 常见错误与避免
One common mistake is forgetting that the centre of enlargement must be specified even when it seems obvious. Another is mixing up the order when calculating the scale factor: scale factor = image length / object length, not the other way round. Students also sometimes add the scale factor instead of multiplying. When the centre is not the origin, they may incorrectly multiply the object coordinates without adjusting for the centre. Practice checking your work by verifying that all rays intersect at a single point.
一个常见错误是，即使放大中心看起来很明显，也忘记必须明确指出。另一个错误是在计算比例因子时搞混顺序：比例因子 = 像的长度 / 原物的长度，切勿反过来。有的同学还会用加法代替乘法。当中心不是原点时，他们可能会错误地直接乘以物体坐标，而没有进行中心调整。养成通过验证所有射线交于一点来检查作业的习惯。
11. Real-life Applications | 实际应用
Enlargements are everywhere: from architects using scale factors to draw building plans, to digital zoom on cameras, and to map reading. When a scale model is built, its dimensions are an enlargement (or reduction) of the real object using a specific scale factor. Even photocopiers allow you to enlarge or reduce a document by a percentage, which is just a scale factor expressed as a percentage. Understanding mathematical enlargements helps you interpret and create accurate proportional representations in design, engineering and everyday life.
放大变换在生活中随处可见：从建筑师使用比例因子绘制建筑平面图，到相机的数码变焦，再到阅读地图。当制作比例模型时，其尺寸就是实物按照特定比例因子的放大（或缩小）。甚至复印机也允许你按百分比放大或缩小文档，这本质上就是用百分数表示的比例因子。理解数学上的放大，能帮助你解读并在设计、工程和日常生活中创建精确的比例表示。
12. Summary and Key Points | 总结与关键点
An enlargement changes the size of a shape, keeping the same proportions. It is defined by a centre and a positive scale factor k. If k > 1, the image is larger; if 0 < k < 1, it is smaller. To fully describe an enlargement, always state 'enlargement', the scale factor and the centre coordinates. Drawing uses the ray method, and coordinate geometry simplifies calculations when the centre is the origin. Mastering this topic builds a bridge to similar triangles, trigonometry and further transformations.
放大改变图形的大小而保持相同比例。它由一个中心和一个正的比例因子 k 来定义。若 k > 1，像变大；若 0 < k < 1，像变小。要完整描述一个放大，必须说明“放大”、比例因子以及中心坐标。绘图使用射线法；当中心为原点时，坐标几何可使计算简化。掌握这一主题将为学习相似三角形、三角学以及更复杂的变换搭建桥梁。
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