Scale factors, angle relationships and triangles
| English | 中文 | Pinyin |
|---|---|---|
| perpendicular height/ˌpɜːpənˈdɪkjʊlə haɪt/ | 垂直高度 | chuí zhí gāo dù |
| similarity/ˌsɪmɪˈlærɪti/ | 相似 | xiāng sì |
A decision before an answer
- A cube with doubled edge has eight times the volume. Doubling its visible width is not doubling every measure.
- Your goal: Apply length, area and volume scaling to similar figures.
Read the relationship
- For similar figures with length scale k, corresponding lengths scale by k, areas by k² and volumes by k³. These laws apply when all corresponding dimensions change together. A cylinder with only radius doubled and fixed height has four times the volume, not eight times. State which dimensions are scaled before choosing the power.
- Use parallel-line angles and triangle sums to find unknowns.
Similar solids have length scale 3. Volume scale:
Volume uses k³.
Use the defining rule
- Use the correct geometric measure: triangle area is half base times perpendicular height; cylinder volume is base area times height. Surface area adds exposed faces, while volume measures enclosed space. A slant length is not automatically the perpendicular height required by an area formula. Attach square or cubic units to the result.
- Distinguish similarity from congruence and adequate information.
Triangle angles 35° and 85° leave:
180-35-85=60.
Check the conditions
- Vertical angles are equal. For parallel lines cut by a transversal, corresponding and alternate interior angles are equal, and same-side interior angles sum to 180°. A triangle’s interior angles sum to 180°; its exterior angle equals the sum of the two remote interior angles. These rules depend on stated or marked parallelism, not the drawing’s appearance.
- Distinguish similarity from congruence and adequate information.
Two similar solids have edge ratio 2:3, so surface-area ratio is 4:9 and volume ratio 8:27. A triangular garden with base 12 m and perpendicular height 5 m has area 30 m². If a triangle has interior angles 48° and 67°, its third is 65° and the adjacent exterior angle is 115°=48°+67°.
Area scale for length scale 4 is ____.
Area uses 4².
Apply the task format
- Similar triangles share corresponding angles and proportional corresponding sides; congruent triangles also have equal corresponding lengths. Two angles establish similarity but not size equality. Maintain correspondence when writing a ratio. A side–side–angle arrangement does not universally guarantee congruence, whereas side–angle–side includes the angle between the two sides.
- Distinguish similarity from congruence and adequate information.
Use k² or k³ only when the entire shape is scaled; do not infer parallel lines or a perpendicular height from an unmarked sketch.
Which answer fits this case?
Apply length, area and volume scaling to similar figures
Two equal corresponding angles prove equal triangle sizes.
They prove similarity, not equal scale.
Keep the distinctions
- similarity 相似 — Matching shape with proportional corresponding lengths.
- perpendicular height 垂直高度 — The shortest perpendicular distance used with a chosen base.
- Apply length, area and volume scaling to similar figures.
- Use parallel-line angles and triangle sums to find unknowns.
- Distinguish similarity from congruence and adequate information.
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.