学习目标
- Recognise, explain, and apply “Ähnlichkeit”.
- Recognise, explain, and apply “Satz des Pythagoras”.
- Recognise, explain, and apply “Satz des Thales”.
- Recognise, explain, and apply “Sinus”.
- Recognise, explain, and apply “Kosinus”.
German Gymnasium Klasse 9 / Mathematics / Curriculum
Geometrie und Trigonometrie
Geometrie und Trigonometrie: structured theory, worked examples, answered practice, and a mastery checklist for German Gymnasium Klasse 9.
章节计划
学习、练习并检查进度
按顺序完成步骤，或直接跳到你需要的部分。
—/5
已完成
预计主动学习时间
209 分钟
1. 理解
学习目标、核心概念与结构化理论。
打开 →2. 掌握方法
逐步说明的完整例题。
打开 →3. 练习
含提示与答案的分级练习。
打开 →4. 巩固
从基础到挑战的课后任务。
打开 →5. 检查
含计时、分值与解答的章节测试。
打开 →
进度只保存在当前浏览器中，不需要 AI credits。
核心概念与检查
- I select a theorem from the diagram's givens.
- I justify every step of the solution.
- Every solution or explanation for “Geometrie und Trigonometrie” should include a method, justification, and final check.
单元
核心理论
练习前按清晰顺序掌握本章核心概念。
Diagram, givens, target
Mark every given on the diagram and state the target clearly. Start from a relationship that connects exactly those quantities.
Property with justification
Every geometric step needs a property, theorem, or definition. A result without justification is not a complete proof.
Units and precision
Length, area, volume, and angle use different units. Round only at the end and keep sufficient intermediate precision.
Ähnlichkeit
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Satz des Pythagoras
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Satz des Thales
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Sinus
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Kosinus
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Mathematics
例题讲解
逐步查看方法，并理解每一步为何成立。
Worked geometry example
A right triangle has perpendicular sides 12 cm and 16 cm. Find its hypotenuse and area.
- 1Pythagoras: c² = 12² + 16².
- 2c = √(400) = 20 cm.
- 3Area = 1/2 · 12 · 16.
Hypotenuse 20 cm, area 96 cm².
Reasoning example
Before calculating, explain the key idea from “Ähnlichkeit” and which conditions must be checked.
- 1Define the idea in one clear sentence.
- 2Connect it to a representation, law, or formula.
- 3State a restriction, unit, or final check that makes the solution valid.
The answer should show not only which rule is used for “Ähnlichkeit”, but also why it is valid here.
Geometrie und Trigonometrie
含答案练习
八道分层练习，从基础熟练到考试应用。请先独立完成，再查看提示或答案。
基础熟练4 分钟
A 7 cm by 6 cm rectangle: find perimeter and area.
提示
P = 2(l+w), A = lw.
答案指南
P = 26 cm, A = 42 cm².
基础熟练5 分钟
A triangle has base 9 cm and height 3 cm. Find its area.
提示
A = 1/2 bh.
答案指南
A = 1/2 · 9 · 3 = 13.5 cm².
应用6 分钟
Two angles in a triangle are 60° and 50°. Find the third.
提示
Triangle angles sum to 180°.
答案指南
70°.
应用7 分钟
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
提示
Use a²+b²=c².
答案指南
10 cm.
推理8 分钟
A circle has radius 4 cm. Find circumference and area in terms of π.
提示
C = 2πr, A = πr².
答案指南
C = 8π cm, A = 16π cm².
推理9 分钟
Scale 1:800. What real length in metres does 6 cm represent?
提示
Multiply in cm, then convert to metres.
答案指南
6 · 800 = 4800 cm = 48 m.
考试题型10 分钟
A cuboid is 4 cm × 3 cm × 7 cm. Find its volume.
提示
V = lwh.
答案指南
V = 84 cm³.
考试题型12 分钟
Points A(0,0) and B(7,5). Find AB.
提示
AB = √(Δx²+Δy²).
答案指南
AB = √74.
常见错误
- Using a formula without checking its conditions.
- Confusing linear, square, and cubic units.
- Treating a not-to-scale diagram as a measurement.
掌握度检查
- I select a theorem from the diagram's givens.
- I justify every step of the solution.
- I use correct units and a final check.
Geometrie und Trigonometrie
章节作业
六道不同作业，从基础到挑战，包含预计用时、提示和答案指南。
基础8 分钟
Homework 1: A 3 cm by 2 cm rectangle: find perimeter and area.
提示
P = 2(l+w), A = lw.
答案指南
P = 10 cm, A = 6 cm².
基础10 分钟
Homework 2: A triangle has base 24 cm and height 4 cm. Find its area.
提示
A = 1/2 bh.
答案指南
A = 1/2 · 24 · 4 = 48 cm².
进阶12 分钟
Homework 3: Two angles in a triangle are 90° and 60°. Find the third.
提示
Triangle angles sum to 180°.
答案指南
30°.
进阶15 分钟
Homework 4: A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
提示
Use a²+b²=c².
答案指南
15 cm.
挑战18 分钟
Homework 5: A circle has radius 5 cm. Find circumference and area in terms of π.
提示
C = 2πr, A = πr².
答案指南
C = 10π cm, A = 25π cm².
挑战20 分钟
Homework 6: Scale 1:400. What real length in metres does 2 cm represent?
提示
Multiply in cm, then convert to metres.
答案指南
2 · 400 = 800 cm = 8 m.
50 分钟 / 40 分
完整章节测试
带计时器、完整分值和参考答案的章节自测。
Start the timer when ready, work without notes, show every step, and open model answers only after finishing.
1. A 8 cm by 2 cm rectangle: find perimeter and area.
2 分答案指南
P = 20 cm, A = 16 cm².
2. A triangle has base 16 cm and height 4 cm. Find its area.
3 分答案指南
A = 1/2 · 16 · 4 = 32 cm².
3. Two angles in a triangle are 70° and 60°. Find the third.
3 分答案指南
50°.
4. A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
4 分答案指南
15 cm.
5. A circle has radius 5 cm. Find circumference and area in terms of π.
4 分答案指南
C = 10π cm, A = 25π cm².
6. Scale 1:900. What real length in metres does 2 cm represent?
4 分答案指南
2 · 900 = 1800 cm = 18 m.
7. A cuboid is 5 cm × 4 cm × 9 cm. Find its volume.
5 分答案指南
V = 180 cm³.
8. Points A(0,0) and B(8,6). Find AB.
5 分答案指南
AB = √100.
9. Similar shapes have length scale factor 1:3. If the small area is 4 cm², find the large area.
5 分答案指南
4 · 3² = 36 cm².
10. Explain which givens and theorem are needed for a complete argument about “Satz des Pythagoras”.
5 分答案指南
Full response: a labelled diagram, givens, an applicable theorem with its conditions, and a logically derived conclusion.
单元
官方来源与核验
课程参考来源；教学顺序请始终与学校和老师确认。
Mathematics
German Gymnasium Klasse 9: 数学与物理的结构化路线，包含章节、练习和下一步。
本章覆盖内容
结构跟随教材，用来帮助组织学习。
Ähnlichkeit
Satz des Pythagoras
Satz des Thales
Sinus
Kosinus
Tangens
Rechtwinklige Dreiecke
Kreisberechnungen
重点关注
通常容易出错或需要更多复习的部分。
I select a theorem from the diagram's givens.
I justify every step of the solution.
I use correct units and a final check.
来源、日常资料与学习资源
从教材、daily material、PDFs、videos 和 worked examples 开始。
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
按小主题练习
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
A 7 cm by 6 cm rectangle: find perimeter and area.
A triangle has base 9 cm and height 3 cm. Find its area.
Two angles in a triangle are 60° and 50°. Find the third.
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
A circle has radius 4 cm. Find circumference and area in terms of π.
Scale 1:800. What real length in metres does 6 cm represent?
A cuboid is 4 cm × 3 cm × 7 cm. Find its volume.
Points A(0,0) and B(7,5). Find AB.
Mocks 与进度检查
如何衡量本章进度，以及何时加入综合 mock。
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
下一步
完成本章后该做什么，以及它如何连接到下一单元。
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
说明：每学年的正式考试范围请始终以学校、老师和当前官方来源为准。