📚 Enlargements: Scale Factors and Centre of Enlargement | 放大：比例因子与放大中心
Enlargement is one of the four key transformations in Key Stage 3 geometry, alongside translation, rotation and reflection. It is the only transformation that changes the size of a shape, making it vital for understanding similarity, scale drawings, maps and even three‑dimensional modelling. This article unpacks everything you need to master enlargements: the role of the scale factor, the importance of the centre of enlargement, how to construct accurate images and how to avoid the most common mistakes.
放大是 KS3 几何中与平移、旋转和反射并列的四种基本变换之一。它是唯一会改变图形大小的变换，因此对于理解相似形、比例图、地图乃至三维建模都至关重要。本文将详细讲解你需掌握的放大知识：比例因子的作用、放大中心的重要性、如何准确作出像以及如何避免最常见的错误。
1. What is an Enlargement? | 什么是放大？
An enlargement is a transformation that changes the size of a shape without altering its proportions. The shape may get larger or smaller, but every angle stays the same and every side keeps the same ratio with the other sides. In technical language, the object and its image are similar figures. Unlike translation, rotation or reflection, enlargement is not a 'rigid motion'; the image is a scaled version of the object.
放大是一种改变图形大小但保持其比例的变换。图形可以变大或变小，但每个角度保持不变，各边之间的比例也保持不变。用术语来说，原像和像是相似图形。与平移、旋转或反射不同，放大不是“刚体运动”；像是原像按比例缩放后的版本。
Every point on the original shape (the object) moves along a straight line that radiates from a fixed point called the centre of enlargement. The distance from the centre to each image point is found by multiplying the distance from the centre to the corresponding object point by a constant — the scale factor. If the scale factor is 2, all distances from the centre double.
原图形（原像）上的每一个点都沿着从固定点（称为放大中心）发出的直线移动。从中心到每个像点的距离，等于从中心到原像对应点距离乘以一个常数——比例因子。如果比例因子为 2，那么从中心出发的所有距离都变为原来的两倍。
2. Scale Factor: The Key to Size Change | 比例因子：尺寸变化的关键
The scale factor (usually written as k) tells you how many times larger or smaller the image will be compared with the object. It applies to every linear measurement: lengths, widths, heights and distances from the centre. A scale factor of 3 means the image is three times as tall and three times as wide as the object.
比例因子（通常记作 k）告诉你像比原像大多少倍或小多少倍。它作用在所有线度上：长、宽、高以及到中心的距离。比例因子为 3 意味着像的高度是原像的三倍，宽度也是原像的三倍。
If k > 1, the image is larger than the object. If 0 < k < 1, the image is smaller — e image is exactly the same size as the object (an identity transformation). Negative scale factors produce an image on the opposite side of the centre, which we will look at later.
如果 k > 1，像比原像大。如果 0 < k < 1，像比原像小——这仍然是放大，但常被称为缩小。如果 k = 1，像与原像全等（恒等变换）。负比例因子会使得像出现在中心的另一侧，我们稍后会讨论。
3. Centre of Enlargement (COE) | 放大中心
The centre of enlargement is the fixed point from which all rays are drawn. It can be anywhere — inside the object, on the boundary, or somewhere outside. When you join corresponding points on the object and image with straight lines, all these lines will meet at the centre of enlargement.
放大中心是所有射线发出的固定点。它可以位于任何位置——原像内部、边界上或外部某处。当你将原像和像上的对应点用直线连接时，所有这些直线会交于放大中心。
If the centre lies on the object, that point does not move during the enlargement. For example, if you enlarge a triangle with the centre at one vertex and a scale factor of 2, that vertex stays in the same place, while the other two vertices move away.
如果中心位于原像上，该点在放大过程中保持不动。例如，若你以一个顶点为放大中心、比例因子为 2 放大一个三角形，该顶点留在原位，而另外两个顶点则向外移动。
4. Positive Scale Factors Greater than 1 | 大于 1 的正比例因子
When the scale factor is greater than 1, the image is larger than the object and lies on the same side of the centre as the object. Each side of the image is k times the length of the corresponding side on the object. The shape remains the same, but all linear dimensions are multiplied by k.
当比例因子大于 1 时，像比原像大，并且位于中心的同侧。像的每条边都是原像对应边长的 k 倍。形状不变，但所有线度都乘以 k。
For instance, take a rectangle with sides 3 cm and 5 cm. Enlarge it with centre O and scale factor 4. The image sides become 12 cm and 20 cm. The image is much larger, but the angles are still 90°. If you measure the distance from O to any image point, it will be exactly 4 times the distance from O to the corresponding object point.
例如，有一个边长为 3 cm 和 5 cm 的长方形。以 O 为中心、比例因子为 4 进行放大。像的边长变为 12 cm 和 20 cm。像大了很多，但角仍然是 90°。如果你测量从 O 到任意像点的距离，它恰好是 O 到对应原像点距离的 4 倍。
5. Scale Factor Between 0 and 1: Reductions | 比例因子在 0 和 1 之间：缩小
A scale factor between 0 and 1 — such as ½ or 0.2 — produces an image that is smaller than the object. The image still lies on the same side of the centre. Each dimension is multiplied by the fraction, so the image shrinks towards the centre.
比例因子在 0 和 1 之间——比如 ½ 或 0.2——产生的像比原像小。像仍然位于中心的同侧。每条边都乘以该分数，使得像向中心缩小。
For example, if a triangle is enlarged with scale factor ⅓, each side of the image is one third of the original side length. The image ends up much closer to the centre than the object. This is useful in creating scale drawings, such as a floor plan where 1 cm represents 3 m in real life — a scale factor of 1/300.
例如，若一个三角形以比例因子 ⅓ 进行放大，像的每条边都是原边长度的三分之一。像最终比原像更靠近中心。这在制作比例图时很有用，例如平面图中 1 cm 代表实际 3 m——相当于比例因子 1/300。
The table below summarises what happens for different positive scale factors k.
下表总结了不同正比例因子 k 的效果。
| Scale Factor (k) | Effect on Image | Position Relative to Centre |
|---|---|---|
| k > 1 | Larger than object | Same side as object |
| k = 1 | Same size as object | Overlaps object |
| 0 < k < 1 | Smaller than object | Same side, closer to centre |
| k = 0 | Image is a single point (the centre) | Centre only |
6. Negative Scale Factors | 负比例因子
A negative scale factor produces an image that is rotated 180° about the centre as well as being enlarged or reduced. The image appears on the opposite side of the centre. If k = –2, the image is twice the size of the object and upside down with respect to the centre.
负比例因子产生的像不仅被放大或缩小，还会绕中心旋转 180°。像出现在中心的另一侧。如果 k = –2，像的大小是原像的两倍，且相对于中心上下颠倒。
To construct an image with a negative scale factor, you still measure the distance from the centre to each object point, multiply by |k|, but then measure that distance in the opposite direction along the same straight line. Rays are extended backwards through the centre.
要作出负比例因子的像，你仍然要测量从中心到每个原像点的距离，乘以 |k|，然后沿着同一条直线向相反方向量取该距离。射线需要反向延伸通过中心。
For k between –1 and 0, the image is a reduction but also inverted. For example, k = – The centre of enlargement always lies between the object and the image when k is negative.
当 k 在 –1 和 0 之间时，像为缩小且倒置。例如，k = –½ 会在另一侧产生大小为原来一半的像。当 k 为负时，放大中心总是位于原像和像之间。
7. How to Construct an Enlargement | 如何作出放大图形
Follow these steps to draw an enlargement accurately:
按照以下步骤准确作出放大图形：
- Locate the centre of enlargement (usually given as a coordinate point).
确定放大中心（通常以坐标点给出）。 - Draw a ray from the centre through a vertex of the object. Repeat for all vertices.
画射线从中心经过原像的一个顶点。对所有顶点重复此操作。 - Measure the distance from the centre to the vertex.
测量从中心到该顶点的距离。 - Multiply this distance by the scale factor. If k is positive, mark the image point on the same ray. If k is negative, mark it on the opposite ray.
将该距离乘以比例因子。如果 k 为正，在射线上同方向标出像点。如果 k 为负，在反向射线上标出。 - Repeat for each vertex and join the image points in the same order to form the image shape.
对每个顶点重复，并按相同顺序连接像点得到像的形状。
Using a ruler and sharp pencil is essential. Many mistakes happen when rays are not drawn precisely or distances are rounded carelessly. Check that all image sides are k times the corresponding object sides.
使用直尺和削尖的铅笔至关重要。许多错误都是由于射线画得不准或距离粗略舍入造成的。检查像的所有边长是否是对应原像边长的 k 倍。
8. Properties of Enlarged Shapes | 放大图形的性质
Enlarged shapes preserve the shape type and all angles. A triangle enlarges to a similar triangle, a square to a square, a circle to a circle. Corresponding angles are equal. Parallel lines in the object remain parallel in the image.
放大后的图形保持图形的种类和所有角度。三角形放大后仍为相似的三角形，正方形仍为正方形，圆仍为圆。对应角相等。原像中的平行线在像中仍保持平行。
The orientation is preserved for positive k, but reversed (rotated 180°) for negative k. The centre of enlargement, object and image are always collinear for each pair of corresponding points.
当 k 为正时，方向不变；当 k 为负时，方向相反（旋转 180°）。对于每一对对应点，放大中心、原像点和像点总是共线的。
ne important property is that the image is not congruent to the object unless k = 1 or k = –1 (the latter gives a 180° rotation, which is an isometry only if |k| = 1). This means areas and volumes change predictably, which we will explore next.
一个重要性质是，除非 k = 1 或 k = –1（后者为旋转 180°，只有当 |k| = 1 时才为等距变换），否则像与原像不全等。这意味着面积和体积会有规律地改变，我们将在下一节探讨。
9. Area and Volume Scale Factors | 面积和体积的比例因子
While the linear scale factor is k, the area scale factor is k², and the volume scale factor is k³. This relationship holds for any 2D shape and any 3D solid under enlargement.
虽然线度比例因子是 k，但面积比例因子是 k²，体积比例因子是 k³。这一关系适用于任何二维图形和任何三维立体在放大变换中的情况。
is very common to see questions asking for the area or volume of an image given the object's measurements and k.
例如，若一个长方形以比例因子 3 放大，其面积变为原来的 3² = 9 倍。若一个立方体以比例因子 2 放大，其体积变为原来的 2³ = 8 倍。考试中经常出现给定原像的测量值和 k，求像的面积或体积的题目。
s ¼ and the volume scale factor is ⅛. So a reduction can dramatically shrink area and volume.
注意：当 k 为 0 和 1 之间的分数时，k² 和 k³ 是更小的分数。例如 k = ½，面积比例因子为 ¼，体积比例因子为 ⅛。因此，缩小会使面积和体积急剧减小。
| Linear Scale Factor (k) | Area Scale Factor | Volume Scale Factor |
|---|---|---|
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| ½ | ¼ | ⅛ |
| –3 | 9 (area is always positive) | –27 (for context, magnitude 27) |
Area scale factor = k² ; Volume scale factor = k³
10. Finding the Centre and Scale Factor from an Enlargement | 由放大图形寻找中心和比例因子
Given an object and its image after an enlargement, you can find both the centre and the scale factor. To locate the centre, draw straight lines through corresponding vertices; the point where all these lines intersect is the centre of enlargement.
给定原像和放大后的像，你可以找到中心和比例因子。要确定中心，画出经过对应顶点的直线；这些直线全部相交的点就是放大中心。
mage point ÷ distance from centre to object point. The two methods should give the same k.
要找出比例因子，选取一对对应边，计算：k = 像的边长 ÷ 原像的边长。或者，测量从中心到原像点和像点的距离：k = 从中心到像点的距离 ÷ 从中心到原像点的距离。两种方法应给出相同的 k。
If the image is on the opposite side of the centre, k is negative. Its magnitude is still the ratio of lengths, but the sign indicates the inversion. Always check the orientation to decide the sign.
如果像在中心的另一侧，则 k 为负。其大小仍然是长度之比，但正负号表示倒置。通常通过检查方向来判断符号。
11. Real-Life Applications | 实际应用
ate floor plans. Photocopiers apply enlargement to zoom in or out of documents, often with scale factors expressed as percentages (e.g. 200% means k = 2).
放大广泛应用于地图绘制，此时远小于 1 的比例因子将实际距离缩小到纸面。建筑师使用缩小来绘制平面图。复印机通过放大缩放文档，通常用百分比表示比例因子（如 200% 表示 k = 2）。
In biology, drawings of cells are often enlarged significantly. A scale bar on a microscope image shows the actual size, and the scale factor can be calculated by comparing the drawn length with the real length. The same principle lies behind blueprints and engineering drawings.
在生物学中，细胞的绘图常常被显著放大。显微镜图像上的比例尺显示实际大小，通过比较绘制长度和实际长度可算出比例因子。同样的原理也应用于蓝图和工程图纸中。
12. Common Mistakes and Tips | 常见错误与提示
Forgetting the centre: Students often try to enlarge by simply multiplying side lengths without using the centre. This produces a shape that is the correct size but in the wrong position. Always draw rays from the centre through each vertex.
忘记放大中心：学生常常只乘以边长而不使用中心。这样画出的图形尺寸正确但位置错误。务必从中心出发经各顶点画射线。
Confusing area and linear scale factors: hat doubling the sides (k = 2) means the area multiplies by 4, not 2. When a question asks 'by what factor does the area increase?', you must square k.
混淆面积比例因子和线度比例因子：记住，边长翻倍（k = 2）意味着面积乘以 4，而非 2。当题目问“面积以什么因子增加？”时，你必须将 k 平方。
Negative scale factor mishandling: For negative k, ensure the image point lies on the opposite ray. A common error is to draw it on the same side with the correct distance, which corresponds to a positive k.
负比例因子处理不当：对于负 k，确保像点位于反向射线上。常见错误是将其画在同侧并取正确距离，而这相当于正 k。
Rays not drawn accurately: Use a sharp pencil and ruler. Even a slight misalignment can make the intersection point for finding the centre unreliable in reverse problems.
射线画得不准确：使用削尖的铅笔和直尺。即使轻微的偏移也可能导致反向找中心时交点不可靠。
Practice with a variety of shapes, centres and scale factors, including fractional and negative values, to build confidence.
通过练习各种图形、中心和比例因子（包括分数和负值）来建立信心。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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