AMC8 2009
AMC8 2009 · Q20
AMC8 2009 · Q20. It mainly tests Combinatorial geometry (counting), Coordinate geometry.
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
如下所示的8个点阵中，有多少个顶点在三个点的非全等的三角形？
(A)
5
5
(B)
6
6
(C)
7
7
(D)
8
8
(E)
9
9
Answer
Correct choice: (D)
正确答案：（D）
Solution
Answer (D): With the points labeled as shown, one set of non-congruent triangles is $AXY$, $AXZ$, $AXW$, $AYZ$, $AYW$, $AZW$, $BXZ$ and $BXW$.
Every other possible triangle is congruent to one of the 8 listed triangles.
CHALLENGE: Find the 48 distinct triangles possible and group them into sets of congruent triangles.
答案（D）：如图所示标记各点时，一组互不全等的三角形为 $AXY$、$AXZ$、$AXW$、$AYZ$、$AYW$、$AZW$、$BXZ$ 和 $BXW$。
其他所有可能的三角形都与上述 8 个三角形中的某一个全等。
挑战：找出所有可能的 48 个不同三角形，并将它们按全等分组。
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