AMC10 2010 A

AMC10 2010 A · Q16

AMC10 2010 A · Q16. It mainly tests Triangles (properties), Divisibility & factors.

Nondegenerate △ABC has integer side lengths, BD is an angle bisector, AD = 3, and DC = 8. What is the smallest possible value of the perimeter?

非退化三角形△ABC具有整数边长，BD是角平分线，AD = 3，DC = 8。周长的最小可能值是多少？

(A) 30 30

(B) 33 33

(D) 36 36

(E) 37 37

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): By the Angle Bisector Theorem, $8 \cdot BA = 3 \cdot BC$. Thus $BA$ must be a multiple of 3. If $BA = 3$, the triangle is degenerate. If $BA = 6$, then $BC = 16$, and the perimeter is $6 + 16 + 11 = 33$.

答案（B）：根据角平分线定理，$8 \cdot BA = 3 \cdot BC$。因此 $BA$ 必须是 3 的倍数。若 $BA = 3$，该三角形为退化三角形。若 $BA = 6$，则 $BC = 16$，周长为 $6 + 16 + 11 = 33$。

Topics