Enlargement and Scale Factor | 放大与比例因子

📚 Enlargement and Scale Factor | 放大与比例因子

Enlargement is one of the four key transformations studied in the Cambridge KS3 mathematics curriculum. It changes the size of a shape but not its proportions, so the image is similar to the original object.

放大变换是剑桥 KS3 数学课程中的四种核心变换之一。它只改变图形的大小而不改变其比例，因此放大后的图形与原图形是相似形。

This revision guide explains how to enlarge shapes, describe enlargements fully, and use scale factors correctly in exam-style questions.

本复习指南将讲解如何放大图形、如何完整描述放大变换，以及如何在考试题型中正确使用比例因子。

1. What Is an Enlargement? | 什么是放大变换？

An enlargement is a transformation that produces an image with exactly the same shape as the original object, but with a different size. All corresponding angles remain unchanged, and every corresponding side is multiplied by the same scale factor.

放大变换是一种产生的图形与原图形形状完全相同、但大小不同的变换。所有对应角保持不变，每条对应边都乘以同一个比例因子。

For example, if a triangle has sides 3 cm, 4 cm and 5 cm, an enlargement by scale factor 2 gives a similar triangle with sides 6 cm, 8 cm and 10 cm.

例如，一个三角形的边长为 3 cm、4 cm 和 5 cm，以比例因子 2 放大后得到的相似三角形边长为 6 cm、8 cm 和 10 cm。

2. Centre and Scale Factor | 中心与比例因子

To describe an enlargement fully, you must state two things: the centre of enlargement and the scale factor, usually written as k. The centre is the fixed point from which all distances are measured.

要完整描述一次放大变换，必须说明两点：放大中心和比例因子，比例因子通常写为 k。放大中心是测量所有距离的固定点。

The scale factor tells you how much the shape is enlarged or reduced. It is the multiplier applied to every distance from the centre to a point on the shape.

比例因子表示图形被放大或缩小的倍数。它是从放大中心到图形上某一点的距离所乘的倍数。

image distance from centre = object distance from centre × k

图像到中心的距离 = 原图形到中心的距离 × k

3. Positive Integer Scale Factors | 正整数比例因子

When the scale factor k is greater than 1, the image is larger than the object. For example, k = 3 means every distance from the centre is multiplied by 3, so the image side lengths are three times the original side lengths.

当比例因子 k 大于 1 时，放大后的图形比原图形大。例如 k = 3 表示从中心到各点的距离都乘以 3，因此图像边长是原边长的三倍。

When k = 1, the image is exactly the same size as the object. This is still an enlargement, but it is an identity transformation because nothing changes.

当 k = 1 时，图像与原图形大小完全相同。这仍然是放大变换，但它是恒等变换，因为图形没有发生变化。

k > 1: the image is larger than the object. 图像比原图形大。
k = 1: the image is the same size as the object. 图像与原图形大小相同。
0 < k < 1: the image is smaller than the object. 图像比原图形小。

4. Drawing an Enlargement Step by Step | 逐步绘制放大图形

To draw an enlargement accurately from a given centre, follow these steps. First, draw straight ray lines from the centre of enlargement through each vertex of the original shape.

要从给定中心精确绘制放大图形，请按照以下步骤操作。首先，从放大中心穿过原图形的每个顶点画直线射线。

Next, measure the distance from the centre to each vertex. Multiply each distance by the scale factor k to find the new distance.

接着，测量从中心到每个顶点的距离。将每个距离乘以比例因子 k，得到新的距离。

Finally, mark each image vertex along the same ray line at the new distance. Join the image vertices to complete the enlarged shape.

最后，沿同一条射线在新的距离处标记每个图像顶点。连接图像顶点，完成放大后的图形。

OA’ = OA × k

新距离 OA’ = 原距离 OA × k

5. Fractional Scale Factors | 分数比例因子

A fractional scale factor between 0 and 1 makes the image smaller than the object. For example, k = ½ means every distance from the centre is halved.

介于 0 和 1 之间的分数比例因子会使图像比原图形小。例如 k = ½ 表示从中心到各点的距离都变为原来的一半。

This is still called an enlargement even though the shape becomes smaller. In examinations, you may see the word ‘reduction’ used informally, but the mathematical transformation is still enlargement.

即使图形变小了，这种变换仍然称为放大变换。在考试中，你可能会看到非正式地使用“缩小”一词，但数学上这种变换仍然是放大。

For instance, a rectangle of length 8 cm and width 4 cm enlarged by scale factor ½ from a centre gives an image of length 4 cm and width 2 cm.

例如，一个长为 8 cm、宽为 4 cm 的矩形以比例因子 ½ 从某个中心放大后，得到长为 4 cm、宽为 2 cm 的图像。

6. Using Coordinates | 使用坐标进行放大

When the centre of enlargement is the origin (0, 0), enlarging a shape is very straightforward. You simply multiply each coordinate of the original vertices by the scale factor k.

当放大中心是原点 (0, 0) 时，放大图形非常简单。你只需将原图形每个顶点的坐标乘以比例因子 k 即可。

If a point has coordinates A(x, y), then its image under enlargement by scale factor k from the origin is A'(kx, ky).

如果某点的坐标为 A(x, y)，那么以原点为中心、比例因子为 k 放大后的图像坐标为 A'(kx, ky)。

Centre (0, 0), k = 2: A(1, 3) becomes A'(2, 6). 中心为 (0, 0)，k = 2：A(1, 3) 变为 A'(2, 6)。
Centre (0, 0), k = ½: B(4, 6) becomes B'(2, 3). 中心为 (0, 0)，k = ½：B(4, 6) 变为 B'(2, 3)。

7. Enlargement on a Grid | 网格上的放大变换

On a coordinate grid or squared paper, ray lines help you check that the centre, the object point and the image point all lie on the same straight line.

在坐标网格或方格纸上，射线可以帮助你检查放大中心、原图形点和图像点是否都在同一条直线上。

If the centre is not at the origin, you can still use the counting method. Count the horizontal and vertical distances from the centre to each vertex, multiply both by k, and then count the same direction to plot the image vertex.

如果中心不在原点，你仍然可以使用数格法。从中心到每个顶点数出水平距离和垂直距离，将两者都乘以 k，然后沿相同方向数格来确定图像顶点。

This method avoids measuring lengths with a ruler and is very useful in grid-based exam questions.

这种方法可以避免用尺子测量长度，在基于网格的考试题中非常有用。

8. Area and Perimeter Effects | 面积与周长变化

When a shape is enlarged by scale factor k, all lengths are multiplied by k. Therefore the perimeter of the image is also multiplied by k.

当图形以比例因子 k 放大时，所有长度都乘以 k。因此图像的周长也乘以 k。

The area of the image is multiplied by k². This is because both length and width are multiplied by k, so the two-dimensional area increases by k × k.

图像的面积则乘以 k²。这是因为长度和宽度都乘以 k，因此二维面积增加了 k × k 倍。

perimeter of image = perimeter of object × k

图像周长 = 原图形周长 × k

area of image = area of object × k²

图像面积 = 原图形面积 × k²

For example, if a square has side length 3 cm and is enlarged by scale factor 4, the new side length is 12 cm and the new area is 144 cm², which is 16 times the original area of 9 cm².

例如，一个边长为 3 cm 的正方形以比例因子 4 放大后，新边长为 12 cm，新面积为 144 cm²，是原面积 9 cm² 的 16 倍。

9. Common Misconceptions | 常见错误

One common mistake is forgetting to give the centre of enlargement when describing a transformation. A full description must include both the centre and the scale factor.

一个常见错误是在描述变换时忘记给出放大中心。完整的描述必须同时包括中心和比例因子。

Another mistake is using the scale factor on side lengths instead of distances from the centre. The enlargement rule is based on distances from the centre, not directly on the original side lengths unless the centre is at a vertex or has a special position.

另一个错误是将比例因子用在边长上，而不是用在从中心出发的距离上。放大规则基于从中心出发的距离，而不是直接基于原边长，除非中心位于某个顶点或有特殊位置。

Students also often confuse area scale factor with length scale factor. Remember: length multiplies by k, but area multiplies by k².

学生也经常混淆面积比例因子和长度比例因子。请记住：长度乘以 k，但面积乘以 k²。

10. Exam-Style Practice Questions | 考试题型练习

Below are three practice questions to test your understanding of enlargement. Try each one before reading the worked hints.

以下是三道练习题，用来检验你对放大变换的理解。在阅读解题提示前，请先尝试独立完成每一道题。

Question 1: A triangle has vertices A(1, 1), B(3, 1) and C(1, 4). Enlarge it by scale factor 3 from the origin. Write down the coordinates of the image vertices. 第 1 题：一个三角形的顶点为 A(1, 1)、B(3, 1) 和 C(1, 4)。以原点为中心、比例因子 3 放大。写出图像顶点的坐标。
Question 2: A rectangle has length 6 cm and width 2 cm. It is enlarged by scale factor ½. Find the perimeter and area of the image. 第 2 题：一个矩形的长为 6 cm、宽为 2 cm。以比例因子 ½ 放大。求图像的周长和面积。
Question 3: Describe fully the enlargement that maps a shape of area 5 cm² onto a similar shape of area 45 cm². 第 3 题：完整描述将面积为 5 cm² 的图形映射到面积为 45 cm² 的相似图形的放大变换。

Hints: For Question 1, multiply each coordinate by 3. For Question 2, the image length is 3 cm and width is 1 cm, so the perimeter is 8 cm and the area is 3 cm². For Question 3, the area scale factor is 45 ÷ 5 = 9, so the length scale factor is √9 = 3.

提示：第 1 题将每个坐标乘以 3。第 2 题图像长为 3 cm、宽为 1 cm，因此周长为 8 cm，面积为 3 cm²。第 3 题面积比例因子为 45 ÷ 5 = 9，因此长度比例因子为 √9 = 3。

11. Summary Checklist | 总结清单

By now you should be able to describe an enlargement fully, draw an enlargement from a given centre, use fractional scale factors, and calculate the effects on perimeter and area.

现在你应该能够完整描述放大变换、从给定中心绘制放大图形、使用分数比例因子，以及计算放大对周长和面积的影响。

An enlargement changes size but not shape. 放大变换只改变大小，不改变形状。
A full description needs centre and scale factor. 完整描述需要中心和比例因子。
Image distance = object distance × k. 图像距离 = 原图形距离 × k。
Area multiplies by k², perimeter by k. 面积乘以 k²，周长乘以 k。
Fractional scale factors less than 1 make the image smaller. 小于 1 的分数比例因子会使图像变小。

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Enlargement and Scale Factor | 放大与比例因子