Similar Figures: Conditions and Applications | 相似图形的判定条件与应用

📚 Similar Figures: Conditions and Applications | 相似图形的判定条件与应用

Similar figures are shapes that have the same shape but not necessarily the same size. In the Edexcel IGCSE Mathematics syllabus, understanding the conditions for similarity and applying them to solve geometric problems is essential. This article will guide you through the key criteria, worked examples, and common applications.

相似图形是指形状相同但大小不一定相同的图形。在爱德思 IGCSE 数学大纲中，理解相似图形的判定条件并运用它们解决几何问题至关重要。本文将带你系统梳理关键判定法则、典型例题和常见应用。

1. Definition of Similar Figures | 相似图形的定义

Two figures are similar if one can be obtained from the other by an enlargement (or a series of enlargements, reflections, rotations, and translations). This means their corresponding angles are equal, and their corresponding sides are in the same ratio.

两个图形相似，是指其中一个图形可以通过放大（或经过放大、反射、旋转和平移的组合）得到另一个图形。这意味着它们的对应角相等，并且对应边成相同的比例。

For example, any two squares are similar because all their angles are 90° and the ratios of corresponding sides are equal. However, two rectangles are not always similar: a 2×3 rectangle and a 4×5 rectangle have equal angles but their side ratios are different (2:3 vs 4:5).

例如，任意两个正方形都相似，因为它们的角都是 90°，且对应边的比例相等。但两个长方形不一定相似：一个 2×3 的长方形和一个 4×5 的长方形角相等，但边比不同（2:3 与 4:5 不同）。

2. Key Conditions for Triangle Similarity | 三角形相似的三大判定条件

For triangles, similarity can be established using three main conditions. You only need one of them to prove that two triangles are similar.

对于三角形，有三大主要判定条件。你只需满足其中一个条件即可证明两个三角形相似。

AA (Angle-Angle): Two pairs of corresponding angles are equal.
SAS (Side-Angle-Side): Two pairs of corresponding sides are in the same ratio, and the included angles are equal.
SSS (Side-Side-Side): All three pairs of corresponding sides are in the same ratio.

AA（两角对应相等）：两组对应角分别相等。
SAS（两边对应成比例且夹角相等）：两组对应边成同一比例，且它们的夹角相等。
SSS（三边对应成比例）：三组对应边都成同一比例。

3. The AA Condition | 两角对应相等（AA）

The AA condition states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. Since the sum of angles in a triangle is always 180°, the third angles must also be equal.

AA 判定条件指出：如果一个三角形的两个角分别等于另一个三角形的两个角，那么这两个三角形相似。由于三角形内角和恒为 180°，第三个角也必然相等。

Consider triangle ABC with ∠A = 50° and ∠B = 60°, and triangle DEF with ∠D = 50° and ∠E = 60°. Then ∠C = 70° and ∠F = 70°, so ΔABC ∼ ΔDEF by AA.

设三角形 ABC 中 ∠A = 50°、∠B = 60°，三角形 DEF 中 ∠D = 50°、∠E = 60°。则 ∠C = 70°、∠F = 70°，因此根据 AA 判定，ΔABC ∼ ΔDEF。

∠A = ∠D 且 ∠B = ∠E ⇒ ΔABC ∼ ΔDEF

In exam questions, look for shared angles, vertically opposite angles, alternate angles (from parallel lines), or angles in the same segment of a circle. These often provide the two equal angle pairs needed for the AA condition.

在考试题中，要善于寻找公共角、对顶角、平行线产生的内错角或同位角，以及同圆中同弧所对的圆周角。这些通常能为你提供 AA 判定所需的两组等角。

4. The SAS Condition | 两边成比例且夹角相等（SAS）

The SAS condition requires two pairs of corresponding sides to be in the same ratio, and the angle between those two sides (the included angle) to be equal in both triangles.

SAS 判定条件要求两组对应边成同一比例，并且这两条边所夹的角（夹角）在两个三角形中相等。

For example, suppose in triangle PQR, PQ = 4 cm, PR = 6 cm, and ∠P = 40°. In triangle XYZ, XY = 6 cm, XZ = 9 cm, and ∠X = 40°. Check the ratio: PQ/XY = 4/6 = 2/3 and PR/XZ = 6/9 = 2/3. The included angles ∠P and ∠X are both 40°, so ΔPQR ∼ ΔXYZ by SAS.

例如，在三角形 PQR 中，PQ = 4 cm，PR = 6 cm，∠P = 40°；在三角形 XYZ 中，XY = 6 cm，XZ = 9 cm，∠X = 40°。检验比例：PQ/XY = 4/6 = 2/3，PR/XZ = 6/9 = 2/3。夹角 ∠P 和 ∠X 均为 40°，因此根据 SAS 判定，ΔPQR ∼ ΔXYZ。

PQ/XY = PR/XZ 且 ∠P = ∠X ⇒ ΔPQR ∼ ΔXYZ

Be careful: the equal angle must be the included angle, i.e., the angle between the two sides you are comparing. If the equal angle is not between the two proportional sides, the condition is not SAS, and the triangles may not be similar.

注意：相等的角必须是夹角，即你所比较的两条边之间的那个角。如果相等的角不是位于两条成比例边的中间，则该条件不构成 SAS，三角形也不一定相似。

5. The SSS Condition | 三边对应成比例（SSS）

The SSS condition states that if all three pairs of corresponding sides are in the same ratio, then the triangles are similar. This is a direct extension of the idea of enlargement.

SSS 判定条件指出：如果三组对应边的长度之比都相同，则这两个三角形相似。这是放大概念的直观延伸。

Suppose triangle ABC has sides AB = 3 cm, BC = 4 cm, CA = 5 cm, and triangle DEF has sides DE = 6 cm, EF = 8 cm, FD = 10 cm. The ratios are 3/6 = 4/8 = 5/10 = 1/2, so ΔABC ∼ ΔDEF by SSS.

设三角形 ABC 的三边为 AB = 3 cm、BC = 4 cm、CA = 5 cm，三角形 DEF 的三边为 DE = 6 cm、EF = 8 cm、FD = 10 cm。比值均为 3/6 = 4/8 = 5/10 = 1/2，因此根据 SSS 判定，ΔABC ∼ ΔDEF。

AB/DE = BC/EF = CA/FD ⇒ ΔABC ∼ ΔDEF

In practice, SSS is less commonly used than AA in IGCSE exams because it requires knowledge of all three side lengths. However, it appears frequently in questions that involve two triangles sharing a common side or where side lengths are given in a diagram.

在实际考试中，SSS 的使用频率低于 AA，因为它需要知道三条边的长度。但在涉及公共边或题目已给出全部边长的图形中，SSS 也很常见。

6. Scale Factor and Enlargement | 比例因子与放大

When two figures are similar, the ratio of any pair of corresponding side lengths is called the linear scale factor (often written as k). If one figure is an enlargement of the other, then every length in the enlarged figure is k times the corresponding length in the original.

当两个图形相似时，任意一组对应边长度的比值称为线性比例因子（通常记作 k）。如果一个图形是另一个图形的放大，那么放大后图形中的每条长度都是原图中对应长度的 k 倍。

For example, if a triangle with sides 5 cm, 6 cm, and 7 cm is enlarged by scale factor 3, the new sides are 15 cm, 18 cm, and 21 cm.

例如，一个三边为 5 cm、6 cm、7 cm 的三角形按比例因子 3 放大后，新边长为 15 cm、18 cm 和 21 cm。

Scale factor k = (new length) ÷ (original length)

If k > 1, the figure is enlarged; if 0 < k < 1, the figure is reduced. When finding a missing side, always identify which triangle or figure is the original and which is the image, then apply the scale factor consistently.

当 k > 1 时，图形被放大；当 0 < k < 1 时，图形被缩小。在求未知边长时，务必先确定哪个是原图形、哪个是变换后的图形，然后一致地应用比例因子。

7. Finding Missing Lengths Using Similarity | 利用相似求未知边长

The most common application of similarity is finding unknown side lengths. Once you have established that two triangles are similar, you can set up a proportion between corresponding sides.

相似最常应用的场景就是求未知边长。一旦确定两个三角形相似，就可以在对应边之间建立比例关系。

Worked example: In triangle ABC, AB = 4 cm, BC = 6 cm, AC = 8 cm. Triangle DEF is similar with DE = 6 cm. Find EF and DF.

例题：在三角形 ABC 中，AB = 4 cm，BC = 6 cm，AC = 8 cm。三角形 DEF 与之相似，且 DE = 6 cm。求 EF 和 DF。

Here AB corresponds to DE. The scale factor is k = DE/AB = 6/4 = 1.5. Therefore EF = 1.5 × BC = 1.5 × 6 = 9 cm, and DF = 1.5 × AC = 1.5 × 8 = 12 cm.

在此，AB 对应 DE。比例因子 k = DE/AB = 6/4 = 1.5。因此 EF = 1.5 × BC = 1.5 × 6 = 9 cm，DF = 1.5 × AC = 1.5 × 8 = 12 cm。

Always state which sides correspond before writing the ratio. A common error is to match sides incorrectly, especially in triangles that have been rotated or reflected. Use the equal angles to identify corresponding vertices.

在列比例式之前，务必先说明哪条边对应哪条边。常见错误是错误匹配边，尤其是在三角形被旋转或反射后。要利用相等的角来确定对应顶点。

8. Similarity in Overlapping Triangles | 重叠三角形中的相似

A classic exam question involves two overlapping triangles with a shared angle. Look for the “A” shape or the “X” shape formed by lines crossing two parallel lines.

经典考题常涉及两个具有公共角的重叠三角形。注意观察由两条平行线被第三条直线所截形成的“A”字形或“X”字形。

In the figure below, DE is parallel to BC in triangle ABC, with D on AB and E on AC.

在下图中，三角形 ABC 中 DE 平行于 BC，点 D 在 AB 上，点 E 在 AC 上。

∠ADE = ∠ABC and ∠AED = ∠ACB ⇒ ΔADE ∼ ΔABC

Since DE ∥ BC, the corresponding angles are equal (alternate angles and common angle at A), so ΔADE ∼ ΔABC by AA. This gives the proportion AD/AB = AE/AC = DE/BC.

因为 DE ∥ BC，对应角相等（同位角以及公共角 ∠A），所以由 AA 判定 ΔADE ∼ ΔABC。由此得到比例 AD/AB = AE/AC = DE/BC。

If AD = 3 cm, DB = 2 cm, and DE = 4 cm, then AB = 5 cm. Thus 4/BC = 3/5, so BC = 4 × 5/3 = 20/3 ≈ 6.67 cm.

若 AD = 3 cm，DB = 2 cm，DE = 4 cm，则 AB = 5 cm。因此 4/BC = 3/5，得 BC = 4 × 5/3 = 20/3 ≈ 6.67 cm。

9. Areas of Similar Figures | 相似图形的面积

If two figures are similar with linear scale factor k, then the ratio of their areas is k². This is because area is a two-dimensional measure: both length and width are multiplied by k.

若两个相似图形的线性比例因子为 k，则它们的面积之比为 k²。这是因为面积是二维度量：长和宽都同时乘以 k。

For example, if a rectangle is enlarged by scale factor 2, its area becomes 4 times larger. If the scale factor is 3, the area becomes 9 times larger.

例如，一个长方形按比例因子 2 放大后，面积变为原来的 4 倍；按比例因子 3 放大后，面积变为原来的 9 倍。

Area scale factor = k²

In IGCSE questions, you may be given the areas of two similar shapes and asked to find the linear scale factor. Remember to take the square root of the area ratio to obtain k.

在 IGCSE 试题中，你可能会已知两个相似图形的面积，要求求出线性比例因子。记住：对面积之比开平方即可得到 k。

For instance, if the areas of two similar triangles are 25 cm² and 100 cm², the area ratio is 100/25 = 4, so the linear scale factor k = √4 = 2. This means each side of the larger triangle is twice the corresponding side of the smaller triangle.

例如，若两个相似三角形的面积分别为 25 cm² 和 100 cm²，则面积之比为 100/25 = 4，因此线性比例因子 k = √4 = 2。这意味着较大三角形的每条边都是较小三角形对应边的 2 倍。

10. Volumes of Similar Solids | 相似立体的体积

For similar 3D solids, the ratio of volumes is the cube of the linear scale factor, k³. Volume is a three-dimensional measure, so each of the three dimensions is multiplied by k.

对于相似的立体图形，体积之比等于线性比例因子的立方，即 k³。体积是三维度量，三个维度都乘以 k。

Suppose two similar cylinders have heights in the ratio 2:3. The linear scale factor is 3/2 = 1.5, so the volume ratio is (1.5)³ = 3.375, or 27:8 if expressed as a ratio.

设两个相似圆柱的高度之比为 2:3。线性比例因子为 3/2 = 1.5，因此体积之比为 (1.5)³ = 3.375，即 27:8。

Volume scale factor = k³

When solving problems, always identify whether the given ratio involves lengths, areas, or volumes. If you are comparing lengths, use k; for areas use k²; for volumes use k³. Converting between these correctly is a key skill in the exam.

解题时，务必先判断题目给出的比值是长度、面积还是体积。比较长度用 k，比较面积用 k²，比较体积用 k³。正确转换这三者是考试中的关键技能。

11. Applications in Real Life | 实际生活中的应用

Similarity is widely used in real-life situations. Architects use scale models to represent buildings; cartographers use similar shapes when creating maps; engineers use similar triangles in measuring heights and distances without direct measurement.

相似在实际生活中的应用非常广泛。建筑师使用比例模型来表现建筑；制图师在制作地图时运用相似图形；工程师利用相似三角形间接测量高度和距离。

One classic application is measuring the height of a tree or building using a shadow. At the same time of day, the sun’s rays make the same angle with the ground, so the tree and its shadow form a triangle similar to the triangle formed by a known vertical object and its shadow.

一个经典应用是利用影子测量树或建筑物的高度。在同一时刻，太阳光线与地面的夹角相同，因此树与它的影子构成的三角形，与一个已知高度的竖直物体及其影子构成的三角形相似。

For example, if a 1.5 m vertical pole casts a shadow of 2 m, and a tree casts a shadow of 10 m at the same time, then the height h of the tree satisfies h/1.5 = 10/2, so h = 1.5 × 5 = 7.5 m.

例如，一根 1.5 m 的竖直杆子在阳光下影长为 2 m，同时一棵树的影长为 10 m，则树高 h 满足 h/1.5 = 10/2，因此 h = 1.5 × 5 = 7.5 m。

This method relies on the AA condition: both the pole and the tree are vertical, so they form right angles with the ground, and the sun’s rays are inclined at the same angle to the ground in both triangles.

这一方法依赖于 AA 判定：杆子和树都垂直于地面，因此与地面成直角，同时太阳光线在两个三角形中与地面的夹角相同。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

Below is a summary of common pitfalls and practical advice for tackling similarity questions in the Edexcel IGCSE exam.

以下总结常见误区以及应对爱德思 IGCSE 考试中相似题目的实用建议。

Misidentifying corresponding sides: always label the similar triangles and match vertices by angle order.
Forgetting to square or cube the scale factor when dealing with areas or volumes.
Using the ASA or AAS condition instead of AA: in IGCSE, we only need AA for similarity, not ASA (the included side is irrelevant for similarity).
Assuming two shapes are similar just because they look alike: you must verify angle equality or side ratio.

错误匹配对应边：务必标记相似三角形，并按照角的顺序匹配顶点。
在涉及面积或体积时忘记对比例因子平方或立方。
误用 ASA 或 AAS 条件：在 IGCSE 中，证明相似只需要 AA，ASA 中的夹边与相似判定无关。
仅凭外形相似就判定两个图形相似：必须验证对应角相等或对应边成比例。

When solving a similarity problem, write the similarity statement first, e.g., ΔABC ∼ ΔDEF. This helps you list corresponding sides correctly and avoids careless errors.

解相似题时，先写出相似关系式，例如 ΔABC ∼ ΔDEF。这有助于你正确列出对应边，避免粗心错误。

Check the units: if lengths are in cm and areas in cm², ensure all conversions are consistent. For volumes, use cm³ or m³ appropriately.

注意单位：长度用 cm，面积用 cm²，体积用 cm³ 或 m³，确保所有换算一致。

In conclusion, mastering the three conditions for triangle similarity — AA, SAS, and SSS — along with the relationships between linear scale factor, area, and volume, will let you solve a wide range of IGCSE problems confidently. Always draw a clear diagram, state the similarity criterion, and set up proportions systematically.

总而言之，掌握三角形相似的三大判定条件——AA、SAS、SSS，以及线性比例因子与面积、体积之间的关系，将使你能够自信地解决各类 IGCSE 题型。绘制清晰图形、明确判定依据、系统建立比例关系，是取得高分的关键。

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Similar Figures: Conditions and Applications | 相似图形的判定条件与应用