Similar Shapes | 相似图形

📚 Similar Shapes | 相似图形

Similar shapes are geometric figures that have exactly the same shape but may differ in size. Understanding the relationships between their lengths, areas and volumes is a core topic in the Edexcel IGCSE Mathematics syllabus.

相似图形是指形状完全相同但大小可能不同的几何图形。理解它们的边长、面积和体积之间的关系是 Edexcel IGCSE 数学大纲的核心内容。

1. What Are Similar Shapes? | 什么是相似图形？

Two shapes are similar if one is an enlargement of the other. This means every corresponding angle is equal, and every corresponding length is multiplied by the same factor, called the scale factor.

如果两个图形中一个是另一个的放大（或缩小），则这两个图形相似。这意味着每个对应角相等，每条对应边都乘以同一个倍数，这个倍数称为比例因子。

For example, any two squares are always similar because all four angles are 90° and the side lengths are always in the same ratio. Likewise, any two circles are similar. However, two rectangles are only similar if their length-to-width ratios are identical.

例如，任意两个正方形总是相似的，因为它们的四个角都是90°，且边长始终成相同比例。同理，任意两个圆也是相似的。但两个矩形只有在长宽比相同时才相似。

Key conditions for similarity:

相似的关键条件：

• Corresponding angles are equal (对应角相等)
• Corresponding sides are in the same ratio (对应边成同一比例)
• One shape is an enlargement or reduction of the other (一个图形是另一个的放大或缩小)

2. The Linear Scale Factor (LSF) | 线性比例因子（LSF）

The linear scale factor (often written as k) is the ratio of a length on the new shape to the corresponding length on the original shape.

线性比例因子（通常写作 k）是新图形中某条边的长度与原图形中对应边长度的比值。

LSF = k = new length ÷ original length

If k > 1, the shape is an enlargement. If 0 < k < 1, the shape is a reduction. If k = 1, the shapes are congruent (identical in size and shape).

如果 k > 1，图形是放大；如果 0 < k < 1，图形是缩小；如果 k = 1，则图形全等（大小和形状完全相同）。

Example: A triangle has sides of 3 cm, 4 cm and 5 cm. A similar triangle has sides of 6 cm, 8 cm and 10 cm.

例：一个三角形的三边为3 cm、4 cm和5 cm。一个相似三角形的三边为6 cm、8 cm和10 cm。

k = 6 ÷ 3 = 8 ÷ 4 = 10 ÷ 5 = 2

The scale factor is 2, meaning every length on the second triangle is twice the corresponding length on the first.

比例因子为2，意味着第二个三角形的每条边都是第一个三角形对应边的两倍。

3. Area Scale Factor (ASF) | 面积比例因子（ASF）

When two shapes are similar with linear scale factor k, the ratio of their areas is k². This is called the area scale factor (ASF).

当两个相似图形的线性比例因子为 k 时，它们的面积之比为 k²。这称为面积比例因子（ASF）。

ASF = k²

Why is this true? Consider a square with side length x. Its area is x². A similar square with side length kx has area (kx)² = k²x². The area is multiplied by k².

为什么是这样？考虑一个边长为 x 的正方形，其面积为 x²。一个边长为 kx 的相似正方形的面积为 (kx)² = k²x²。面积被乘以 k²。

Example: Two similar rectangles have a linear scale factor of 3. The smaller rectangle has an area of 5 cm².

例：两个相似矩形的线性比例因子为3

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