AMC10 2006 B

AMC10 2006 B · Q15

AMC10 2006 B · Q15. It mainly tests Triangles (properties), Similarity.

Rhombus ABCD is similar to rhombus BFDE. The area of rhombus ABCD is 24, and ∠BAD = 60°. What is the area of rhombus BFDE?

菱形 ABCD 与菱形 BFDE 相似。菱形 ABCD 的面积为 24，∠BAD = 60°。求菱形 BFDE 的面积。

(A) 6 6

(B) 4√3 4√3

(D) 9 9

(E) 6√3 6√3

Answer

Correct choice: (C)

正确答案：（C）

Solution

Since ∠BAD = 60°, isosceles △BAD is also equilateral. As a consequence, △AEB, △AED, △BED, △BFD, △BFC, and △CFD are congruent. These six triangles have equal areas and their union forms rhombus ABCD, so each has area 24/6 = 4. Rhombus BFDE is the union of △BED and △BFD, so its area is 8.

因为 ∠BAD = 60°，等腰 △BAD 也是等边三角形。因此 △AEB、△AED、△BED、△BFD、△BFC 和 △CFD 全等。这六个三角形面积相等，它们的并构成菱形 ABCD，故每个面积为 24/6 = 4。菱形 BFDE 是 △BED 和 △BFD 的并，故面积为 8。

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