Similar Triangles: Proofs and Applications | 相似三角形：证明与应用

📚 Similar Triangles: Proofs and Applications | 相似三角形：证明与应用

Similar triangles are one of the most powerful tools in IGCSE Mathematics. Once you understand how to prove two triangles are similar, you unlock a whole range of problem-solving strategies involving lengths, areas, and real-world measurements.

相似三角形是 IGCSE 数学中最强大的工具之一。一旦你理解了如何证明两个三角形相似，你就解锁了一系列涉及长度、面积和现实世界测量的解题策略。

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1. What Does “Similar” Mean? | 什么是“相似”？

Two triangles are said to be similar if they have exactly the same shape but not necessarily the same size. This means their corresponding angles are equal and their corresponding sides are in the same ratio (also called the scale factor).

如果两个三角形形状完全相同但大小不一定相同，则称它们为相似三角形。这意味着它们对应的角相等，且对应的边成相同的比例（也称为比例因子或缩放因子）。

Mathematically, if triangle ABC is similar to triangle DEF, we write:

在数学上，如果三角形 ABC 相似于三角形 DEF，我们记作：

△ABC ∼ △DEF

This notation implies that A corresponds to D, B corresponds to E, and C corresponds to F.

这种记法意味着 A 对应 D，B 对应 E，C 对应 F。

• Corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
• 对应角相等：∠A = ∠D, ∠B = ∠E, ∠C = ∠F
• Corresponding sides are proportional: AB/DE = BC/EF = AC/DF
• 对应边成比例：AB/DE = BC/EF = AC/DF

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2. The Four Tests for Similarity | 相似三角形的四种判定方法

In IGCSE, you must know four ways to prove that two triangles are similar. They are abbreviated as AA, SAS, SSS, and RHS.

在 IGCSE 考试中，你必须掌握四种证明两个三角形相似的方法。它们缩写为 AA、SAS、SSS 和 RHS。

Test | 判定方法	Condition | 条件
AA (Angle-Angle) | 两角对应相等	Two angles of one triangle equal two angles of the other. | 一个三角形的两个角与另一个三角形的两个角对应相等。
SAS (Side-Angle-Side) | 两边成比例且夹角相等	Two sides are in the same ratio and the included angle is equal. | 两边对应成比例，且它们的夹角相等。
SSS (Side-Side-Side) | 三边对应成比例	All three pairs of corresponding sides are in the same ratio. | 三对对应边的比例都相同。
RHS (Right angle-Hypotenuse-Side) | 直角、斜边和一条直角边	Both triangles are right-angled, with hypotenuse and one side in the same ratio. | 两个三角形都是直角三角形，且斜边和一条直角边对应成比例。

Note that AA is the most commonly used test because in many geometry problems, angles are easier to identify than side ratios.

注意，AA 是最常用的判定方法，因为在许多几何问题中，找角比找边的比例更容易。

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3. The AA Test in Action | AA 判定法的实际应用

Consider two triangles ABC and DEF. If ∠A = ∠D = 40° and ∠B = ∠E = 65°, then automatically ∠C = ∠F = 75° because the angle sum of a triangle is always 180°. Therefore, △ABC ∼ △DEF by AA.

考虑两个三角形 ABC 和 DEF。如果 ∠A = ∠D = 40° 且 ∠B = ∠E = 65°，那么根据三角形内角和为 180°，自动得出 ∠C = ∠F = 75°。因此，根据 AA 判定法，△ABC ∼ △DEF。

Here are the most common angle relationships that establish equal angles:

以下是建立相等角的常见角度关系：

• Vertically opposite angles are equal. | 对顶角相等。
• Alternate angles on parallel lines are equal. | 平行线上的内错角相等。
• Corresponding angles on parallel lines are equal. | 平行线上的同位角相等。
• Angles in the same segment of a circle are equal. | 圆中同弧上的圆周角相等。

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4. The SAS and SSS Tests | SAS 和 SSS 判定法

Use the SAS test when you know two sides of each triangle and the angle between those two sides. For example, if AB/DE = AC/DF = 2 and ∠A = ∠D, then the triangles are similar with a scale factor of 2.

当你已知每个三角形的两条边以及这两条边之间的夹角时，使用 SAS 判定法。例如，如果 AB/DE = AC/DF = 2 且 ∠A = ∠D，则这两个三角形相似，比例因子为 2。

Use the SSS test when you know all three sides of both triangles. Simply check that all three ratios are equal. For instance, if the sides of one triangle are 3 cm, 4 cm, and 5 cm, and the sides of another are 6 cm, 8 cm, and 10 cm, then the ratio is 1:2 for all sides, so the triangles are similar.

当你已知两个三角形的三条边时，使用 SSS 判定法。只需验证三个比例是否都相等。例如，如果一个三角形的三边为 3 cm、4 cm 和 5 cm，而另一个三角形的三边为 6 cm、8 cm 和 10 cm，则所有边的比例都是 1:2，因此这两个三角形相似。

Important: In SSS similarity, the sides must be listed in the correct order. The longest side of one triangle must correspond to the longest side of the other, and so on.

重要提示：在用 SSS 判定相似时，边的对应顺序必须正确。一个三角形的最长边必须对应另一个三角形的最长边，以此类推。

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5. Similarity vs. Congruence | 相似与全等的区别

Many students confuse similar triangles with congruent triangles. Let us clarify the difference once and for all.

许多学生容易混淆相似三角形与全等三角形。让我们彻底厘清它们的区别。

Property | 性质	Similar Triangles | 相似三角形	Congruent Triangles | 全等三角形
Shape | 形状	Same | 相同	Same | 相同
Size | 大小	May differ | 可能不同	Same | 相同
Corresponding angles | 对应角	Equal | 相等	Equal | 相等
Corresponding sides | 对应边	In proportion | 成比例	Equal | 相等

In fact, congruence is simply similarity with a scale factor of exactly 1. Every congruent pair is also similar, but not every similar pair is congruent.

事实上，全等就是比例因子恰好为 1 的相似。每一对全等三角形也一定是相似的，但并非每一对相似三角形都全等。

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6. Finding Unknown Side Lengths | 求未知边长

The most common exam question involving similar triangles asks you to find an unknown side length. Once you have established that two triangles are similar, you simply set up a proportion.

涉及相似三角形最常见的考题是求未知边长。一旦你确认了两个三角形相似，只需建立比例关系即可。

Suppose △ABC ∼ △DEF, with AB = 4 cm, BC = 6 cm, AC = 5 cm, and DE = 10 cm. To find EF, write:

假设 △ABC ∼ △DEF，其中 AB = 4 cm，BC = 6 cm，AC = 5 cm，DE = 10 cm。要求 EF，可列式：

BC/EF = AB/DE

6/EF = 4/10

Cross-multiplying gives 4 × EF = 60, so EF = 15 cm.

交叉相乘得到 4 × EF = 60，因此 EF = 15 cm。

Step-by-step process:

解题步骤：

• Identify the corresponding sides. | 找出对应边。
• Write a proportion using the known sides. | 用已知边列出比例式。
• Cross-multiply and solve for the unknown. | 交叉相乘并解出未知量。
• Check that the answer is reasonable. | 检查答案是否合理。

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7. Scale Factors: Linear, Area, and Volume | 比例因子：线性、面积和体积

When two shapes are similar with a linear scale factor k, their areas scale by k² and their volumes scale by k³. This pattern appears frequently in IGCSE papers.

当两个图形相似且线性比例因子为 k 时，它们的面积按 k² 缩放，体积按 k³ 缩放。这一规律在 IGCSE 试卷中频繁出现。

For example, if the sides of two similar triangles are in the ratio 2:3, then:

例如，如果两个相似三角形的边长比为 2:3，那么：

• Linear scale factor k = 3/2 | 线性比例因子 k = 3/2
• Area ratio = k² = 9/4 | 面积比 = k² = 9/4

Area of larger triangle / Area of smaller triangle = (3/2)² = 9/4

If the smaller triangle has an area of 8 cm², the larger one has area 8 × 9/4 = 18 cm².

如果较小三角形的面积为 8 cm²，则较大三角形的面积为 8 × 9/4 = 18 cm²。

For volume, the same logic extends: k³. If you double all lengths, volume increases by 2³ = 8 times.

对于体积，同样的逻辑扩展为 k³。如果将所有长度加倍，体积增大 2³ = 8 倍。

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8. Using Similar Triangles to Prove Geometric Theorems | 用相似三角形证明几何定理

Similar triangles are often used to prove important geometric results. One classic example is proving that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side.

相似三角形常用于证明重要的几何结论。一个经典的例子是证明连接三角形两边中点的线段平行于第三边。

Take triangle ABC, with M the midpoint of AB and N the midpoint of AC. Since AM/AB = AN/AC = 1/2 and ∠A is common, we have △AMN ∼ △ABC by SAS. Therefore ∠AMN = ∠ABC, which means MN is parallel to BC. This is a beautiful and concise proof.

设三角形 ABC 中，M 是 AB 的中点，N 是 AC 的中点。因为 AM/AB = AN/AC = 1/2，且 ∠A 为公共角，根据 SAS 判定法，△AMN ∼ △ABC。因此 ∠AMN = ∠ABC，这意味着 MN 平行于 BC。这是一个优美而简洁的证明。

Another important application is in proving the intercept theorem (Thales’ theorem), which deals with parallel lines cutting transversals in equal ratios.

另一个重要的应用是证明截线定理（泰勒斯定理），该定理涉及平行线按相等比例截割截线。

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9. Word Problems and Real-World Applications | 应用题与真实世界中的应用

Similar triangles are widely used in real life. One of the most common examples is using shadows to measure the height of a tall object. This method is often called the “shadow method” and dates back to ancient Greek mathematics.

相似三角形在现实生活中应用广泛。最常见的例子之一是使用影子来测量高大物体的高度。这种方法常被称为“影子法”，其历史可追溯到古希腊数学。

Example: A vertical flagpole casts a shadow of 12 m. At the same time, a 1.5 m tall person standing nearby casts a shadow of 2 m. How tall is the flagpole?

例题：一根垂直旗杆的影子长 12 m。与此同时，一个身高 1.5 m 的人站在旁边，影子长 2 m。旗杆有多高？

Since the sun’s rays are parallel, the two triangles formed are similar:

由于太阳光线是平行的，两个三角形构成相似三角形：

Height of flagpole / Height of person = Shadow of flagpole / Shadow of person

x / 1.5 = 12 / 2

Solving: x = 1.5 × 6 = 9 m. The flagpole is 9 metres tall.

解得：x = 1.5 × 6 = 9 m。旗杆高 9 米。

Other applications include map reading (where distances are scaled), designing models of buildings, and calculating distances across rivers or lakes that cannot be measured directly.

其他应用包括地图判读（距离按比例缩放）、建筑设计模型，以及计算无法直接测量的河流或湖泊两岸的距离。

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10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Through years of marking past papers, examiners find that students repeatedly make the same mistakes. Here are the most common ones and how to avoid them.

通过多年批改真题，考官发现学生会反复犯相同的错误。以下是最常见的错误及避免方法。

• Matching the wrong sides: Always carefully match sides that are opposite equal angles. Draw the triangles separately if necessary.
• 错误匹配边：始终仔细匹配相等角所对的边。如有必要，将两个三角形分开画。
• Using sides that are not corresponding: In a proportion equation, both numerators must come from the same triangle.
• 使用不对应的边：在比例方程中，两个分子必须来自同一个三角形。
• Forgetting to state the similarity reason: In exam questions that ask for proof, you must clearly state which test (AA, SAS, SSS, RHS) you are using.
• 忘记说明相似的理由：在要求证明的考题中，你必须清楚地说明你使用的是哪种判定方法（AA、SAS、SSS、RHS）。
• Confusing area scale factor with linear scale factor: Remember that area multiplies by k², not by k.
• 混淆面积比例因子与线性比例因子：记住面积乘以 k²，而不是 k。

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11. Exam Tips and Worked Examples | 考试技巧与例题精讲

Here is a typical IGCSE exam question that combines several skills.

以下是一道结合了多种技能的典型 IGCSE 考题。

Question: In triangle ABC, point D lies on AB such that AD = 4 cm and DB = 6 cm. Point E lies on AC such that DE is parallel to BC. If AE = 5 cm, find EC.

题目：在三角形 ABC 中，点 D 在 AB 上，AD = 4 cm，DB = 6 cm。点 E 在 AC 上，且 DE 平行于 BC。若 AE = 5 cm，求 EC。

Solution:

解答：

Since DE is parallel to BC, we know that ∠ADE = ∠ABC (corresponding angles) and ∠AED = ∠ACB (corresponding angles). Therefore, by AA, △ADE ∼ △ABC.

因为 DE 平行于 BC，所以 ∠ADE = ∠ABC（同位角）且 ∠AED = ∠ACB（同位角）。因此，根据 AA 判定法，△ADE ∼ △ABC。

Now we set up the proportion using the side lengths along AB:

现在我们利用 AB 上的边长建立比例式：

AD/AB = AE/AC

We know AD = 4 and AB = AD + DB = 4 + 6 = 10. Therefore:

已知 AD = 4，AB = AD + DB = 4 + 6 = 10。因此：

4/10 = 5/AC

Cross-multiplying: 4 × AC = 50, so AC = 12.5 cm. Thus EC = AC − AE = 12.5 − 5 = 7.5 cm.

交叉相乘：4 × AC = 50，所以 AC = 12.5 cm。因此 EC = AC − AE = 12.5 − 5 = 7.5 cm。

Always remember to check whether the question asks for EC or AC. Many students accidentally give the value of AC instead of EC.

始终注意题目要求的是 EC 还是 AC。许多学生不小心给出了 AC 的值而不是 EC。

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12. Summary and Final Checklist | 总结与最终自检清单

Similar triangles are a core topic in IGCSE Mathematics. Mastery of this topic earns marks not only in geometry questions but also in trigonometry, mensuration, and problem-solving contexts.

相似三角形是 IGCSE 数学的核心知识点。掌握该主题不仅能在几何题中得分，还能在三角函数、度量以及应用题中得分。

Before you move on, make sure you can confidently do the following:

在继续之前，请确保你能自信地完成以下内容：

• State all four tests of similarity (AA, SAS, SSS, RHS). | 准确陈述四种相似判定方法（AA、SAS、SSS、RHS）。
• Prove that two triangles are similar using angle and/or side relationships. | 利用角或边的关系证明两个三角形相似。
• Find unknown lengths by setting up correct proportions. | 通过建立正确的比例式求未知长度。
• Apply the relationships between linear, area, and volume scale factors. | 应用线性、面积和体积比例因子之间的关系。
• Solve word problems using similar triangles. | 用相似三角形解决应用题。

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