Circle theorems and reasoned proofs · Higher · 圆的定理与推理证明 · 高阶
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| cyclic quadrilateral/ˈsaɪklɪk ˌkwɒdrɪˈlætərəl/ | 圆内接四边形 | yuán nèi jiē sì biān xíng |
Which angles see the same chord?
- Two observers see the same chord from the circle. Their angles are linked by a theorem rather than by the apparent size of the drawing.
- This lesson studies cyclic quadrilateral 圆内接四边形: A quadrilateral whose four vertices lie on one circle.
Choose the mathematical structure
- The angle at the centre is twice the angle at the circumference on the same arc. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral sum to 180°. A radius is perpendicular to a tangent at contact.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines cyclic quadrilateral? · 以下哪项描述正确定义了圆内接四边形？
A quadrilateral whose four vertices lie on one circle. · 四个顶点共圆的四边形。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
If a central angle is 100°, the corresponding angle at the circumference is 50°. In a cyclic quadrilateral with one angle 112°, its opposite angle is 180-112=68°. A radius meeting a tangent gives 90°, even if the drawing looks oblique.
Circle theorems and reasoned proofs · 圆的定理与推理证明
The angle at the centre is twice the angle at the circumference on the same arc · 同弧所对的圆心角是圆周角的两倍
Choose the theorem from the geometric configuration before calculating the angle. · 根据几何构型选择相应的定理后再计算角度。
Find the circumference angle corresponding to a 100° central angle. · 求对应于100°圆心角的圆周角。
For the same chord and corresponding arc, centre angle=twice circumference angle: 100/2=50°. · 对于同一条弦及其对应的弧，圆心角等于圆周角的两倍：100/2=50°。
Test a tempting shortcut
- Identify the same chord and the correct arc before using a theorem. Two visible right angles do not prove a quadrilateral cyclic without a valid converse argument. A diagram need not be to scale.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every quadrilateral has opposite angles summing to 180°. This claim is false. Explain which definition or assumption it violates.
Find the angle opposite 112° in a cyclic quadrilateral. · 求圆内接四边形中与112°相对的角。
Opposite cyclic angles sum to 180°, so 180-112=68°. · 圆内接四边形的对角之和为180°，因此180-112=68°。
Every quadrilateral has opposite angles summing to 180°. · 任意四边形的对角之和均为180°。
Identify the same chord and the correct arc before using a theorem. Two visible right angles do not prove a quadrilateral cyclic without a valid converse argument. A diagram need not be to scale. · 使用定理前请先识别同一条弦及正确的弧。仅凭图中可见的两个直角无法证明四边形是圆内接四边形，除非有有效的逆命题论证。图形不必按比例绘制。
Interpret a new situation
- Write one reason alongside each angle calculation. For the alternate-segment theorem, name the tangent and chord, then identify the angle in the opposite segment. Use auxiliary radii only when they help the proof.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the angle between a radius and its tangent in degrees. · 求半径与其切线之间的夹角（以度为单位）。
A radius and its tangent are perpendicular at contact: 90°. · 半径与切线在接触点处垂直：90°。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A quadrilateral whose four vertices lie on one circle. Choose the relationship, show the method, check its assumptions and interpret the result.