AMC10 2011 B

AMC10 2011 B · Q18

AMC10 2011 B · Q18. It mainly tests Angle chasing, Triangles (properties).

Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that ∠AMD = ∠CMD. What is the degree measure of ∠AMD ?

矩形 ABCD 有 AB = 6，BC = 3。在边 AB 上选择点 M，使得 ∠AMD = ∠CMD。∠AMD 的度数是多少？

(A) 15 15

(B) 30 30

(D) 60 60

(E) 75 75

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): Sides AB and CD are parallel, so ∠CDM = ∠AMD. Because ∠AMD = ∠CMD, it follows that △CMD is isosceles and CD = CM = 6. Therefore △MCB is a 30–60–90° right triangle with ∠BMC = 30°. Finally, 2 · ∠AMD + 30°= ∠AMD + ∠CMD + 30°= 180°, so ∠AMD = 75°.

答案 (E)：边 AB 和 CD 平行，所以 ∠CDM = ∠AMD。因为 ∠AMD = ∠CMD，故 △CMD 是等腰三角形，CD = CM = 6。因此 △MCB 是 30–60–90° 直角三角形，∠BMC = 30°。最后，2 · ∠AMD + 30°= ∠AMD + ∠CMD + 30°= 180°，所以 ∠AMD = 75°。

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