Geometry and Trigonometry
| English | 中文 | Pinyin |
|---|---|---|
| similarity/ˌsɪmɪˈlærɪti/ | 相似 | xiāng sì |
| radian/ˈreɪdɪən/ | 弧度 | hú dù |
A decision before an answer
- A map doubles every length. The represented area grows by four, not two.
- Your goal: Use area, volume and similarity.
Read the relationship
- Similarity preserves corresponding angles and length ratios. Areas scale by k² and volumes by k³.
- Apply right-triangle ratios and the Pythagorean theorem.
A circle has equation (x−2)²+(y+3)²=25. Centre?
Compare the signs with (x−h)²+(y−k)².
Use the defining rule
- For a right triangle, sin is opposite/hypotenuse, cos adjacent/hypotenuse, and tan opposite/adjacent.
- Use circle equations, arcs and angle relationships.
A cube’s edge doubles. Its volume grows by:
Three independent lengths each double: 2³=8.
Check the conditions
- A circle with centre (h,k) and radius r has (x−h)²+(y−k)²=r².
- Use circle equations, arcs and angle relationships.
A right triangle has legs 6 and 8. c²=6²+8²=100, so c=10. For the angle opposite 6, sin θ=6/10=0.6. A similar triangle enlarged by factor 3 has area nine times the original.
For radius 3 and angle 2 radians, arc length is ____.
s=rθ=3×2=6 in the radius unit.
Apply the task format
- A full turn is 360° or 2π radians. Arc length is rθ only when θ is in radians.
- Use circle equations, arcs and angle relationships.
Radius and diameter differ by a factor of two. Using diameter in πr² makes the area four times too large.
Which answer fits this case?
Use area, volume and similarity
sin θ can exceed 1 in a real right triangle.
The opposite side cannot exceed the hypotenuse.
Keep the distinctions
- similarity 相似 — Equal angles and proportional corresponding lengths.
- radian 弧度 — Angle measure defined by arc length divided by radius.
- Use area, volume and similarity.
- Apply right-triangle ratios and the Pythagorean theorem.
- Use circle equations, arcs and angle relationships.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.