גיאומטריה: עובדות מסומנות, קואורדינטות, מעגלים וצורות תלת-ממדיות
הוצג| English | משמעות |
|---|---|
| scale factor/skeɪl ˈfæktə/ | גורם קנה מידה |
| sector/ˈsektə/ | קטע עיגול |
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:
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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:
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√(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.
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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.
איזו תשובה מתאימה למקרה זה?
שימוש בקשרים הנתונים ולא בקנה מידה apparent של ציור
A triangle drawn almost equilateral must have three equal sides.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.