📚 Similar Shapes | 相似形
In geometry, two shapes are 'similar' if they have the same shape, but not necessarily the same size. Understanding similar shapes is essential for solving real-world problems involving scale models, maps, and enlargements. In IGCSE Mathematics, you will learn how to use scale factors for lengths, areas, and volumes to find missing measurements and solve multi-step problems.
在几何中，如果两个图形形状相同但大小不一定相同，它们就是“相似”的。理解相似形对于解决涉及比例模型、地图和放大的实际问题至关重要。在IGCSE数学中，你将学习如何使用长度、面积和体积的比例因子来找到缺失的测量值，并解决多步骤问题。
1. What are Similar Shapes? | 什么是相似形？
sponding sides are constant. For example, if triangle ABC is similar to triangle PQR, then ∠A = ∠P, ∠B = ∠Q, ∠C = ∠, and AB/PQ = BC/QR = CA/RP = k, where k is the scale factor. The symbol for similarity is '∼'.
如果两个图形中一个是另一个的放大，则它们相似。这意味着所有对应角相等且对应边的比例恒定。例如，如果三角形ABC相似于三角形PQR，则∠A = ∠P，∠B = ∠Q，∠C = ∠R，且AB/PQ = BC/QR = CA/RP = k，其中k是比例因子。相似符号为'∼'。
Similar shapes can appear in any position: they may be rotated, reflected or translated, but these transformations do not change the shape – only the size differs by a constant scale factor.
相似图形可以出现在任何位置：它们可能被旋转、反射或平移，但这些变换不会改变形状——只有大小按恒定的比例因子变化。
2. Conditions for Similarity in Triangles | 三角形的相似条件
ides are in the same proportion. If AB/PQ = BC/QR = CA/RP, then the triangles are similar.
对于三角形，我们可以使用三个主要条件来证明相似。SSS（边边边）：所有三组对应边成相同比例。如果AB/PQ = BC/QR = CA/RP，则三角形相似。
es are in proportion and the included angles are equal. For instance, if AB/PQ = AC/PR and ∠A = ∠P, then △ABC ∼ △PQR.
SAS（边角边）：两组对应边成比例且夹角相等。例如，如果AB/PQ = AC/PR且∠A = ∠P，则△ABC ∼ △PQR。
AA (Angle-Angle): two pairs of corresponding angles are equal. Because the sum of angles in a triangle is 180°, the third pair must also be equal. This is often the quickest condition to use.
AA（角角）：两对对应角相等。由于三角形内角和为180°，第三对角必然也相等。这通常是最快捷的判定方法。
3. Finding the Length Scale Factor | 求长度比例因子
in image / length in original. If k > 1, the image is an enlargement; if 0 < k < 1, it is a reduction.
要确定两个相似形的比例因子，将像（新图形）的边长除以原图形的对应边长。比例因子 k = 像中的长度 / 原图中的长度。如果 k > 1，像是放大；如果 0 < k < 1，则是缩小。
6 cm. A similar larger rectangle has length 15 cm. The length scale factor k = 15/6 = 2.5.
例如，一个小矩形长6 cm。一个相似的大矩形长15 cm。长度比例因子 k = 15/6 = 2.5。
4. Calculating Unknown Lengths | 计算未知长度
Once the scale factor k is known, any unknown length in the similar shape can be found by multiplying the corresponding original length by k. Always check that you are matching corresponding sides correctly – drawing and labelling both shapes is a useful strategy.
一旦知道了比例因子k，相似图形中的任何未知长度都可以通过将对应原长度乘以k来求得。务必确保正确匹配对应边——画出并标注两个图形是一个有用的策略。
The width of the similar larger rectangle is 4 × 2.5 = 10 cm. If instead the larger shape' th is given as 12 cm, you can find k by 12/4 = 3, and then use that to find other lengths.
用之前的矩形例子：原矩形的宽是4 cm。相似大矩形的宽为 4 × 2.5 = 10 cm。如果反过来给出大矩形的宽为12 cm，你可以用12/4 = 3求出k，然后用它找出其他长度。
5. Area Scale Factor | 面积比例因子
When two shapes are similar, the ratio of their areas is equal to the square of the length scale factor. If the length scale factor is k, then the area scale factor is k². This rule applies to all types of areas: surface area of 3D objects, area of circles, triangles, and any other 2D shapes.
当两个图形相似时，它们的面积比等于长度比例因子的平方。如果长度比例因子为k，则面积比例因子为k²。此规则适用于所有类型的面积：3D物体的表面积、圆的面积、三角形的面积以及任何其他2D图形。
the smaller triangle is 5 cm². The area of the larger triangle is 5 × 2² = 5 × 4 = 20 cm².
例子：两个相似三角形的对应边分别为3 cm和6 cm，因此k = 6/3 = 2。小三角形面积为5 cm²，大三角形面积为5 × 2² = 5 × 4 = 20 cm²。
instance, if two similar shapes have areas 25 cm² and 100 cm², the area ratio is 4, so k = √4 = 2.
如果给你的是面积比，取平方根即可求得k。例如，如果两个相似图形的面积分别为25 cm²和100 cm²，面积比为4，因此 k = √4 = 2。
6. Volume Scale Factor | 体积比例因子
ace area still uses k². This applies to cylinders, cuboids, spheres, and any solid where all dimensions are scaled by the same factor.
对于相似的3D图形，体积比等于长度比例因子的立方。体积比例因子 = k³。表面积仍使用k²。这适用于圆柱体、长方体、球体以及所有尺寸按相同比例缩放立体图形。
the smaller cone is 10 cm³. The volume of the larger cone is 10 × 3³ = 10 × 27 = 270 cm³.
例子：两个相似圆锥的高度分别为4 cm和12 cm，因此k = 12/4 = 3。小圆锥体积为10 cm³，大圆锥体积为10 × 3³ = 10 × 27 = 270 cm³。
If you know the volume ratio V, find k by taking the cube root: k = ∛V. For example, if the volumes are 8 m³ and 64 m³, the ratio is 8, so k = ∛8 = 2.
如果你知道体积比V，通过对V开立方根求得k：k = ∛V。例如，如果体积为8 m³和64 m³，比值为8，因此k = ∛8 = 2。
7. Similar Triangles in Real-World Problems | 实际问题中的相似三角形
Similar triangles often appear in questions involving shadows, maps, or overlapping figures. When two triangles share an angle or have parallel sides, look for equal corresponding angles to establish AA similarity. A common problem: finding the height of an object using its shadow and the shadow of a known reference.
相似三角形经常出现在涉及影子、地图或重叠图形的问题中。当两个三角形共角或有平行边时，寻找相等的对应角来建立AA相似性。一个常见问题：利用物体影子与已知参照物的影子来求物体的高度。
shadow 2.4 m long. A flagpole casts a shadow 9.6 m long at the same time. Because the sun' ow is constant: height of flagpole / 9.6 = 1.8 / 2.4, so height = 9.6 × (1.8/2.4) = 7.2 m.
例子：一个身高1.8 m的人影长为2.4 m。同一时刻，一根旗杆的影长为9.6 m。由于太阳光线平行，三角形相似。高度与影长之比恒定：旗杆高度 / 9.6 = 1.8 / 2.4，因此高度 = 9.6 × (1.8/2.4) = 7.2 m。
8. Multi-step Problems with Scale Factors | 比例因子的多步骤问题
In exam questions, you may be given the area or volume ratio and asked to find an unknown length. The key is to first identify what type of ratio you have and then convert it to the length scale factor. The table below summarises the conversions.
在考试问题中，你可能会被给出面积比或体积比，要求求出某个未知长度。关键是首先识别你拥有的是哪种比例，然后将其转换为长度比例因子。下表总结了转换方法。
| Given Ratio | Length SF (k) | Area SF | Volume SF |
|---|---|---|---|
| Length SF = k | k | k² | k³ |
| Area SF = A | √A | A | (√A)³ |
| Volume SF = V | ∛V | (∛V)² | V |
eight of 8 cm. Find the height of the larger vase. Step 1: Area ratio = 200/50 = 4, so k =
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