ت.9 · التشابه، التوسع ومواقع المسافة المخروطية
GRE Subject Test · GRE Mathematics الموضوع 22 3:07 سرد باللغة الإنجليزية · ترجمة مدمجة بالإنجليزية + الصينية
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This unit matches triangle vertices by equal angles before forming a side ratio, applies length, area and volume scale factors, and identifies conic distance conditions while keeping a signed hyperbola branch.
本单元在构造边长比之前先按等角匹配三角形的顶点，应用长度、面积和体积的比例因子，并识别圆锥曲线的距离条件，同时保留带符号的双曲线分支。
Similar triangles have equal corresponding angles and equal ratios of corresponding sides.
相似三角形的对应角相等，对应边之比相等。
Write the vertex correspondence explicitly: if angle P equals angle C, then P matches C and Q matches B.
要明确写出顶点的对应关系：若 ∠P = ∠C，则 P 对应 C，Q 对应 B。
AP lies on AB but corresponds to AC.
AP 在 AB 上，但对应的是 AC。
With a length scale factor k, lengths multiply by k, areas by k squared and volumes by k cubed.
长度比例因子为 k 时，长度乘以 k，面积乘以 k²，体积乘以 k³。
A 30 percent length increase means k equals 1.30, giving an area increase of 69 percent, not 60.
长度增加 30% 即 k = 1.30，面积增加 69%，不是 60%。
Similar solids with volume ratio 27 have length ratio 3 and area ratio 9.
体积比为 27 的相似立体，长度比为 3，面积比为 9。
A circle fixes distance to one point; an ellipse fixes the sum of distances to two foci; a hyperbola fixes the absolute difference; a parabola equates distance to a focus and to a directrix.
圆固定到一点的距离；椭圆固定到两焦点的距离之和；双曲线固定距离之差的绝对值；抛物线使到焦点的距离等于到准线的距离。
A signed difference picks one hyperbola branch, and squaring can lose that sign.
带符号的差选出双曲线的一支，而平方可能丢失这个符号。
AP over AC is 6 over 15, two fifths, so PQ is 8 and AQ is 4.
AP/AC = 6/15 = 2/5，所以 PQ = 8，AQ = 4。
Separately, with foci at minus 5 and 5 and distance-to-left minus distance-to-right equal to 6, a is 3 and b squared is 16.
另外，焦点在 (±5, 0)，到左焦点距离减到右焦点距离等于 6，则 a = 3，b² = 16。
The positive difference selects the right branch: at 3, 0 the distances 8 and 2 differ by 6; at minus 3, 0 the difference is minus 6, and the left branch fails.
正的差选出右支：在 (3, 0)，距离 8 和 2 相差 6；在 (−3, 0)，差为 −6，左支不满足。
Match angles before sides.
先匹配角，再匹配边。
Area percentages use the square of the length factor.
面积百分比要用长度因子的平方。
A squared distance equation may describe both branches even when the problem asks for only one.
平方后的距离方程可能描述两支，即使题目只要其中一支。
Taking positive square roots gives length ratio 3 to 5, and cubing gives volume ratio 27 to 125.
取正平方根得长度比 3 : 5，立方得体积比 27 : 125。
The area ratio is not itself a length scale.
面积比本身不是长度比例。
Similarity is assumed here; area ratios alone do not prove it.
这里相似是前提；仅凭面积比不能证明相似。
The squares of the two distances differ by 20x, so their sum is 10x over 3, and r B is 5x over 3 minus 3, which must be nonnegative.
两个距离的平方差为 20x，所以两距离之和是 10x/3，r_B = 5x/3 − 3，且必须非负。
Squaring and collecting gives x squared over 9 minus y squared over 16 equals 1 on the right branch.
平方并整理得 x²/9 − y²/16 = 1，取右支。
A difference of 12 exceeds the focal distance 10: impossible.
差为 12 超过焦距 10：不可能。
Check yourself.
自我检测。
A 30 percent length increase raises area by 69 percent.
长度增加 30%，面积增加 69%。
Volume ratio 27 means length ratio 3.
体积比 27 意味着长度比 3。
Area 9 to 25 gives volume 27 to 125.
面积比 9 : 25 给出体积比 27 : 125。
The positive difference picks the branch x at least 3.
正的差选出 x ≥ 3 这一支。
And a difference larger than the focal distance is impossible.
大于焦距的差是不可能的。