Similarity, scaling and conic distance loci
| English | 中文 | Pinyin |
|---|---|---|
| focus/ˈfəʊkəs/ | 焦点 | jiāo diǎn |
| similarity/ˌsɪmɪˈlærɪti/ | 相似 | xiāng sì |
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?
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?
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 ____.
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.
Which answer fits this case?
Match triangle vertices by equal angles before forming a side ratio
The signed condition distance to (−5,0) minus distance to (5,0) equals 6 includes both hyperbola branches.
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.
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.