Geometry: labelled facts, coordinates, circles and solids
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| scale factor/skeɪl ˈfæktə/ | 缩放比例 | suō fàng bǐ lì |
| sector/ˈsektə/ | 扇形 | shàn xíng |
A decision before an answer
- A picture that looks like a square is not evidence of equal sides or right angles unless the question establishes them.
- Your goal: Use given relationships rather than a diagram’s apparent scale.
Read the relationship
- Extract the facts a diagram supplies: lengths, right-angle marks, parallel lines, circle radii and stated shape properties. Avoid measuring the screen. Angle sums and similarity give relationships when their conditions hold. Pythagorean reasoning needs a right triangle; it does not apply to every drawn triangle.
- Connect coordinate distances, slopes and geometric properties.
A similar figure has lengths doubled. Its area multiplies by:
Area uses the square of the linear factor.
Use the defining rule
- For two coordinates, use midpoint ((x₁+x₂)/2,(y₁+y₂)/2) and distance √((x₂−x₁)²+(y₂−y₁)²). Slope is Δy/Δx when Δx≠0. A vertical line has no finite slope. Combine coordinate and shape facts where useful; perpendicular nonvertical lines have slopes whose product is −1.
- Distinguish linear, area and volume scale factors.
Distance from (0,0) to (5,12) is:
√(25+144)=13.
Check the conditions
- A circle has circumference 2πr and area πr². An arc or sector uses its fraction of a full turn. Rectangular solids use volume lwh; cylinders use πr²h. Surface area counts exposed faces or surfaces, not the enclosed volume. Attach units so an area answer is not confused with a length or volume.
- Distinguish linear, area and volume scale factors.
Points A(−1,2) and B(5,10) have Δx=6, Δy=8. Distance d=√(6²+8²)=10 and midpoint M=((−1+5)/2,(2+10)/2)=(2,6). For a circle of radius 6, a 60° sector is 1/6 of a full circle: area=(60/360)π6²=6π; arc length=(60/360)2π6=2π. Enlarging every length by 3 multiplies area by 9 and volume by 27, not 3.
A cylinder with r=2 and h=3 has volume ____π.
V=πr²h=π·4·3.
Apply the task format
- For similar figures with linear factor k, corresponding areas multiply by k² and volumes by k³. If a question provides only an area ratio, take a square root to recover the length ratio; do not copy the area ratio directly. General-test geometry uses elementary relationships, with no trigonometry or calculus required for this lesson’s tasks.
- Distinguish linear, area and volume scale factors.
Use labelled facts, include required π or units, and distinguish a perimeter from an area or a sector from its arc.
Which answer fits this case?
Use given relationships rather than a diagram’s apparent scale
A triangle drawn almost equilateral must have three equal sides.
Drawing appearance does not establish equal sides.
Keep the distinctions
- sector 扇形 — A circle region bounded by two radii and their included arc.
- scale factor 缩放比例 — The multiplier relating corresponding lengths of similar figures.
- Use given relationships rather than a diagram’s apparent scale.
- Connect coordinate distances, slopes and geometric properties.
- Distinguish linear, area and volume scale factors.
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.