Angle reasoning, similarity and mensuration · Extension · 角度推理、相似性与度量 · 拓展
| 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.
体积是否随长度同比例缩放？
- 某模型的尺寸是实物的三分之一。其表面积和体积分别缩小了多少倍？
- 本课学习相似比（scale factor）：关联相似图形对应边长的倍数。
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.
选择数学结构
- 对于长度相似比为 k 的相似图形，面积按 k² 缩放，体积按 k³ 缩放。需明确陈述角度相关的几何理由。圆的切线垂直于过切点的半径。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
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.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
若模型到实际长度的比例因子为3，则面积为12 cm²的模型对应实际面积为12×3²=108 cm²；体积为8 cm³的模型对应实际体积为8×3³=216 cm³。一个半径r=3、高h=5的圆柱体，其体积为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.
检验一个诱人的捷径
- 仅凭对应角相等只能判定相似，不能判定全等。请使用顺序对应的边长。在计算面积或体积前，先将长度单位换算一致，或将换算因子的平方/立方正确应用于面积/体积计算。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
将实体的所有长度加倍，其体积也会加倍。此说法是错误的。请说明它违背了哪个定义或假设。
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.
解读新情境
- 几何证明应注明相关定理，指出相等的角或比值，并得出结论。只有当题目允许测量时，按比例绘制的图才可作为证据。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
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
- 9260 · Extension · 3.3. 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.
The multiplier that relates corresponding lengths in similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- 9260 · 拓展内容 · 3.3。在分配拓展内容前，请先匹配目标层级和课程大纲要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
关联相似图形对应边长的倍数。请选择关系式，展示计算步骤，检查其适用假设并解读结果。