유사성, 스케일링 및 원뿔 거리 궤적
도입됨| English | 의미 |
|---|---|
| focus/ˈfəʊkəs/ | 초점 |
| similarity/ˌsɪmɪˈlærɪti/ | 유사성 |
A decision before an answer
- A map enlarges every distance by 30%. The inked area grows by 69%, while a signed difference of distances to two landmarks can select only one branch of a curve.
- Your goal: Match triangle vertices by equal angles before forming a side ratio.
Read the relationship
- Triangles are similar when their corresponding angles agree; their corresponding side ratios are then equal. Write the vertex correspondence explicitly. If triangle APQ has angle A equal to angle A of ABC and angle P equal to angle C, its ordered correspondence is A↔A, P↔C, Q↔B. Thus AP/AC=AQ/AB=PQ/CB. The side AP lies on AB but corresponds to AC; matching by where a side lies or by the letter P can give the wrong ratio.
- Apply length, area and volume scale factors to geometric changes.
Correspondence is A↔A, P↔C, Q↔B. If AP=6, AC=15 and CB=20, what is PQ?
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AP corresponds to AC, not AB. The common scale is 6/15=2/5; multiply CB=20 by it.
Use the defining rule
- With a common length scale factor k>0, lengths multiply by k, areas by k² and volumes by k³. A 30% length increase means k=1.30, hence an area increase of 1.30²−1=0.69, or 69%; it is not 60%. For similar solids with a volume ratio of 27, the length ratio is 3 and the surface-area ratio 9. Percentage changes and absolute differences are different quantities; write the new-to-old ratio before converting it to a percent.
- Identify conic distance conditions and retain a signed hyperbola branch.
A circle radius grows by 30%. What is the area increase?
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The new area is 1.3²=1.69 times the old area, so the increase is 69%, not the full new ratio 169%.
Check the conditions
- A circle fixes distance to one point. An ellipse fixes the sum of distances to two foci. A hyperbola fixes the absolute difference of those distances; a parabola fixes equality of distance to a focus and to a directrix. These are definitions of point sets, not sketches to memorise. For foci (−c,0) and (c,0), an ellipse uses a sum 2a with a>c. A nondegenerate hyperbola uses an absolute difference 2a with 0<a<c; a signed difference distinguishes its two branches.
- Identify conic distance conditions and retain a signed hyperbola branch.
Let AB=10, AC=15 and BC=20. A point P on AB has AP=6, and Q on AC makes angle APQ equal to angle ACB. Then AP/AC=6/15=2/5, so PQ=8 and AQ=4. Separately, foci at (−5,0),(5,0) and distance-to-left minus distance-to-right equal to 6 give a=3, b²=25−9=16: x²/9−y²/16=1 with x≥3. The point (−3,0) satisfies the squared equation but fails the signed condition.
For foci (−5,0),(5,0) and a hyperbola difference magnitude 6, the value of b² is ____.
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c=5 and a=3; b²=c²−a²=25−9=16.
Apply the task format
- For the hyperbola with horizontal transverse axis, c²=a²+b² and the equation is x²/a²−y²/b²=1. Let r_A be distance to (−c,0) and r_B distance to (c,0). The condition r_A−r_B=2a>0 selects the right branch x≥a; the negative condition selects the left. Squaring can lose this sign, so check the original distance condition after obtaining the equation. Difference zero gives the perpendicular bisector rather than a hyperbola; a difference exceeding the focal separation is impossible by the triangle inequality.
- Identify conic distance conditions and retain a signed hyperbola branch.
Match angles before sides. Area percentages use the square of the length factor. A squared distance equation may describe both branches even when the original problem asks for only one.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
변장비율을 형성하기 전에 동일한 각도를 가진 삼각형의 꼭짓점을 매칭함
The signed condition distance to (−5,0) minus distance to (5,0) equals 6 includes both hyperbola branches.
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The positive signed difference selects the right branch. At the left vertex the difference is −6.
Keep the distinctions
- similarity 相似 — Equality of corresponding angles and a common ratio of corresponding lengths.
- focus 焦点 — A fixed point used in a conic distance definition.
- Match triangle vertices by equal angles before forming a side ratio.
- Apply length, area and volume scale factors to geometric changes.
- Identify conic distance conditions and retain a signed hyperbola branch.
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.