Angle reasoning, similarity and mensuration · 角度推理、相似性与度量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| scale factor/skeɪl ˈfæktə/ | 相似比 | xiāng sì bǐ |
Does volume scale like length?
- A model has lengths one third of the real object. How much smaller are its area and volume?
- This lesson studies scale factor 相似比: The multiplier that relates corresponding lengths in similar shapes.
Choose the mathematical structure
- For similar shapes with length scale factor k, areas scale by k² and volumes by k³. State angle reasons explicitly. A circle's tangent is perpendicular to the radius at the contact point.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines scale factor? · 以下哪项描述正确定义了比例因子？
The multiplier that relates corresponding lengths in similar shapes. · 关联相似图形对应长度的乘数。
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 model-to-real length factor is 3, a model area of 12 cm² gives 12×3²=108 cm² and a model volume of 8 cm³ gives 8×3³=216 cm³. A cylinder with r=3,h=5 has volume πr²h=45π.
Angle reasoning, similarity and mensuration · 角度推理、相似性与度量
For similar shapes with length scale factor k, areas scale by k² and volumes by k³ · 对于长度比例因子为 k 的相似图形，面积按 k² 缩放，体积按 k³ 缩放。
Match the scale factors to the correct dimension before using 12 cm² or 8 cm³. · 在使用 12 cm² 或 8 cm³ 之前，请将比例因子与正确的维度进行匹配。
Two similar figures have length factor 3. Find the area factor. · 两个相似图形的长度因子为 3。求面积因子。
Area scales by the square of the length factor: 3²=9. · 面积按长度因子的平方缩放：3²=9。
Test a tempting shortcut
- Equal angles alone establish similarity, not equal size. Use corresponding lengths in the same order. Convert linear units before calculating area or volume, or square/cube the conversion factor correctly.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Doubling every length of a solid doubles its volume. This claim is false. Explain which definition or assumption it violates.
Two similar solids have length factor 3. Find the volume factor. · 两个相似立体的长度因子为 3。求体积因子。
Volume scales by the cube: 3³=27. · 体积按立方缩放：3³=27。
Doubling every length of a solid doubles its volume. · 将立体的所有边长加倍，其体积也会加倍。
Equal angles alone establish similarity, not equal size. Use corresponding lengths in the same order. Convert linear units before calculating area or volume, or square/cube the conversion factor correctly. · 仅相等的角只能确定相似性，不能确定大小。需按相同顺序使用对应边长。在计算面积或体积前应先转换线性单位，或正确对换算因子进行平方/立方运算。
Interpret a new situation
- A geometric proof should name the relevant theorem, identify the equal angle or ratio, and draw the conclusion. A scale drawing is evidence only when the task permits measurement.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the sum of interior angles of a pentagon in degrees. · 求五边形内角和（单位：度）。
Interior-angle sum=(5-2)×180=540°. · 内角和 = (5 - 2) × 180 = 540°。
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
The multiplier that relates corresponding lengths in similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.