Angle reasoning, similarity and mensuration · Extension · Raisonnement angulaire, similarité et mensuration · Élargissement
| English | Français |
|---|---|
| scale factor/skeɪl ˈfæktə/ | facteur d'échelle |
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? · Quelle description définit correctement le facteur d'échelle ?
The multiplier that relates corresponding lengths in similar shapes. · Le multiplicateur reliant les longueurs correspondantes dans des figures similaires.
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 · Raisonnement sur les angles, similarité et mensuration
For similar shapes with length scale factor k, areas scale by k² and volumes by k³ · Pour des formes semblables avec un facteur d'échelle linéaire k, les aires sont multipliées par k² et les volumes par k³
Match the scale factors to the correct dimension before using 12 cm² or 8 cm³. · Associez les facteurs d'échelle à la bonne dimension avant d'utiliser 12 cm² ou 8 cm³.
Two similar figures have length factor 3. Find the area factor. · Deux figures similaires ont un facteur de longueur 3. Trouver le facteur d'aire.
Area scales by the square of the length factor: 3²=9. · L'aire est proportionnelle au carré du facteur de longueur : 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. · Deux solides semblables ont un facteur de longueur 3. Trouvez le facteur de volume.
Volume scales by the cube: 3³=27. · Le volume est proportionnel au cube : 3³=27.
Doubling every length of a solid doubles its volume. · Doublez chaque longueur d'un solide, son volume double également.
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. · Des angles égaux seuls établissent la similarité, pas l'égalité de taille. Utiliser les longueurs correspondantes dans le même ordre. Convertir les unités linéaires avant de calculer l'aire ou le volume, ou carré/cube correctement le facteur de conversion.
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. · Trouver la somme des angles intérieurs d'un pentagone en degrés.
Interior-angle sum=(5-2)×180=540°. · Somme des angles intérieurs=(5-2)×180=540°.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
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.