📚 Enlargement and Similarity: Scale Factor, Centre and Similar Triangles | 放大与相似形：比例因子、放大中心与相似三角形
An enlargement is a transformation that changes the size of a shape but keeps its angles and overall proportions unchanged. In Cambridge KS3 Mathematics, students learn to enlarge shapes using a scale factor and a centre of enlargement, then move on to identify similar shapes and solve problems involving coordinates and area.
放大是一种图形变换，它改变图形的大小，但保持图形的角和整体比例不变。在剑桥 KS3 数学中，学生需要学习如何使用比例因子和放大中心来放大图形，进而识别相似形，并解决涉及坐标和面积的问题。
1. What is an enlargement? | 什么是放大？
An enlargement changes all side lengths of a shape by the same multiplier, called the scale factor. The shape still looks the same, but it is bigger or smaller. A true enlargement preserves angle sizes, so the original shape and the enlarged shape are always similar.
放大按照同一个倍数来改变图形所有边的长度，这个倍数就是比例因子。放大后的图形看起来仍与原图形一样，只是变大或变小了。真正的放大保持角的大小不变，因此原图形与放大后的图形始终是相似的。
Key features of an enlargement include a centre of enlargement, a scale factor, and corresponding points connected by straight lines passing through the centre.
放大包含三个关键要素：放大中心、比例因子，以及对应点之间经过放大中心的直线。
2. Scale factor | 比例因子
The scale factor tells us how many times longer each side of the image is compared with the original. If the scale factor is k, then every image length is k × the corresponding original length.
比例因子告诉我们放大后图形的每一条边是原图形对应边的多少倍。如果比例因子是 k，那么放大后每一条边的长度就是原对应边长度的 k 倍。
image length = k × original length
放大后边长 = k × 原边长
For example, if k = 3, all lengths in the image are three times the matching lengths in the original shape. This relationship must be applied to every side.
例如，如果 k = 3，放大后图形的所有长度都是原图形对应长度的三倍。这种关系必须应用于每一条边。
3. Centre of enlargement | 放大中心
The centre of enlargement is a fixed point from which enlargement rays are drawn. Each point on the original shape moves along a ray starting at the centre. The distance from the centre to the image point is k × the distance from the centre to the original point.
放大中心是一个固定点，放大射线从该点发出。原图形上的每个点都沿着从中心出发的射线移动。中心到放大后对应点的距离，是中心到原图形对应点距离的 k 倍。
To enlarge a shape accurately, first draw rays from the centre through each vertex. Then measure the distance from the centre to each vertex, multiply it by k, and plot the new image point on the same ray.
要准确地放大一个图形，首先从中心经过每个顶点画射线。然后测量中心到每个顶点的距离，乘以 k，并在同一条射线上标出新的像点。
4. Finding the centre of enlargement | 寻找放大中心
When an enlargement has already been drawn, the centre can be found by joining corresponding points on the original and the image. The centre is the point where these rays intersect.
当放大图形已经画出时，可以通过连接原图形和放大后图形上的对应点来找到放大中心。放大中心就是这些射线相交的点。
- Draw a straight line through a point and its image point.
- Draw another straight line through a second point and its image point.
- The intersection of these lines is the centre of enlargement.
- 画一条直线连接一个点和它的像点。
- 再画一条直线连接另一个点和它的像点。
- 这两条直线的交点就是放大中心。
This method works because every enlargement keeps the centre, the original point and the image point collinear.
这种方法之所以有效，是因为在放大中，放大中心、原图形点和它的像点始终在同一条直线上。
5. Positive scale factors greater than 1 | 大于1的比例因子
When k is greater than 1, the image is larger than the original. For example, if k = 2, each side of the image is twice as long as the corresponding side of the original. The shape is enlarged and appears further from the centre.
当比例因子 k 大于 1 时，放大后的图形比原图形大。例如，如果 k = 2，放大后图形每条边的长度都是原图形对应边的两倍。图形被放大，并且距离放大中心更远。
In coordinate problems, a point with coordinates (x, y) enlarged by factor 2 from the origin becomes (2x, 2y). This only applies when the centre is the origin.
在坐标问题中，以原点为放大中心、比例因子为 2 时，点 (x, y) 会变为 (2x, 2y)。这种规则仅适用于放大中心是原点的情况。
6. Fractional scale factors (reduction) | 分数比例因子（缩小）
A fractional scale factor between 0 and 1 makes the image smaller than the original. For example, if k = 1/2, every side of the image is half the length of the corresponding side of the original. This transformation is sometimes called a reduction.
比例因子介于 0 和 1 之间的分数会使放大后的图形比原图形小。例如，如果 k = 1/2，放大后图形每条边的长度都是原图形对应边的一半。这种变换有时称为缩小。
When the centre is the origin, enlarging (x, y) by factor 1/2 gives (x/2, y/2). The image moves closer to the centre but remains the same shape.
当放大中心为原点时，以比例因子 1/2 放大点 (x, y) 会得到 (x/2, y/2)。像点会靠近放大中心，但形状保持不变。
7. Negative scale factors | 负比例因子
A negative scale factor produces an image on the opposite side of the centre of enlargement. The shape is also inverted. For example, if k = -1, the image is the same size as the original but rotated by 180 degrees around the centre.
负比例因子会使放大后的图形出现在放大中心的另一侧，图形还会上下颠倒。例如，如果 k = -1，放大后的图形与原图形大小相同，但绕放大中心旋转了 180 度。
For k = -2, the image is twice as large as the original, on the opposite side of the centre. Distance from the centre is multiplied by 2, but direction is reversed.
对于 k = -2，放大后的图形是原图形的两倍大，并位于放大中心的另一侧。到中心的距离会乘以 2，但方向相反。
8. Similar shapes and similarity | 相似形与相似性
Two shapes are similar if one is an enlargement of the other. This means corresponding angles are equal and corresponding sides are in the same ratio. Any enlargement produces a similar shape.
如果一个图形是另一个图形的放大，那么这两个图形相似。这意味着对应角相等，对应边成相同比例。任何放大都会产生相似形。
In similar triangles, if the sides of the smaller triangle are a, b and c, and the scale factor is k, then the sides of the larger triangle are ka, kb and kc. The angles remain unchanged.
在相似三角形中，如果较小三角形的边长是 a、b 和 c，比例因子是 k，那么较大三角形的边长就是 ka、kb 和 kc。角度保持不变。
9. Enlargement on coordinate grids | 坐标平面上的放大
Coordinate grids are useful for testing enlargement skills. When the centre is the origin, each coordinate is simply multiplied by the scale factor. If point A is (3, 4) and k = 2, the image A’ is (6, 8).
坐标平面非常适合考查放大知识。当放大中心是原点时，每个坐标只需要乘以比例因子。如果点 A 是 (3, 4)，比例因子 k = 2，那么像点 A’ 就是 (6, 8)。
If the centre is not the origin, first find the vector from the centre to the point, multiply the vector by k, then add this result back to the centre coordinates.
如果放大中心不是原点，首先找到从放大中心到该点的向量，将向量乘以 k，然后再将结果加回放大中心的坐标。
image point = centre + k × (original point – centre)
像点 = 放大中心 + k × (原图形点 – 放大中心)
10. Area and scale factor | 面积与比例因子
When a shape is enlarged by a length scale factor k, its area is multiplied by k². This is because both the base and the height of the shape are multiplied by k.
当一个图形按长度比例因子 k 放大时，它的面积会乘以 k²。这是因为图形的底和高都乘了 k。
new area = k² × original area
新面积 = k² × 原面积
For example, if k = 3, the new area is 9 times the original area. Understanding this relationship helps solve problems involving area changes in enlargements and similar shapes.
例如，如果 k = 3，新面积就是原面积的 9 倍。理解这一关系有助于解决涉及放大和相似形面积变化的问题。
11. Exam-style questions and tips | 考试题型与技巧
Cambridge KS3 questions often ask students to enlarge a shape from a given centre, describe an enlargement, find the scale factor, or calculate the area after enlargement. Marks are awarded for correct image position, accurate side lengths and correct labelling.
剑桥 KS3 考题经常要求学生从给定中心放大图形、描述一次放大、求出比例因子，或计算放大后的面积。评分标准通常关注像点位置是否正确、边长是否准确以及标记是否正确。
- Always draw rays from the centre through each vertex before plotting the image.
- Label the image points clearly, for example A’, B’, C’.
- Check that corresponding sides are parallel when the scale factor is positive.
- Use squared paper to keep lines accurate and measurements consistent.
- 在画像点之前，一定要从放大中心经过每个顶点画射线。
- 清楚地标记像点，例如 A’、B’、C’。
- 当比例因子为正时，检查对应边是否平行。
- 使用方格纸保持线条准确和测量一致。
12. Summary | 总结
Enlargement is a core Cambridge KS3 transformation that prepares students for similarity and scale factor work. Remember that the scale factor multiplies all lengths, the centre of enlargement controls direction, and the area changes by k². Accurate ray drawing and coordinate methods are essential for exam success.
放大是剑桥 KS3 数学中重要的图形变换，为学生进一步学习相似形和比例因子打下基础。请记住：比例因子会乘以所有长度，放大中心控制方向，面积则按 k² 变化。准确的射线作图和坐标方法是考试成功的关键。
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