AMC12 2025 B

AMC12 2025 B · Q10

AMC12 2025 B · Q10. It mainly tests Triangles (properties), Ratios in geometry.

The altitude to the hypotenuse of a $30^\circ{-}60^\circ{-}90^\circ$ is divided into two segments of lengths $x<y$ by the median to the shortest side of the triangle. What is the ratio $\tfrac{x}{x+y}$?

$30^\circ{-}60^\circ{-}90^\circ$ 三角形的斜边上的高被到最短边的中线分成两段 $x<y$。求 $\tfrac{x}{x+y}$ 的值。

(A) \dfrac{3}{7} \dfrac{3}{7}

(B) \dfrac{\sqrt3}{4} \dfrac{\sqrt3}{4}

(D) \dfrac{5}{11} \dfrac{5}{11}

(E) \dfrac{4\sqrt3}{15} \dfrac{4\sqrt3}{15}

Answer

Correct choice: (A)

正确答案：（A）

Solution

Without loss of generality, let $\triangle ABC$ have side-lengths $AB=2, BC=2\sqrt{3},$ and $AC=4.$ Let $D$ be the foot of the perpendicular from $B$ to $\overline{AC}, \ E$ be the midpoint of $\overline{AB},$ and $F$ be the intersection of $\overline{CE}$ and $\overline{BD}.$ Note that $\triangle ADB$ and $\triangle BDC$ are both $30^\circ{-}60^\circ{-}90^\circ$ triangles. From the side-length ratio, we get $AD=1$ and $DC=3.$ We obtain the following diagram: From here, we will proceed with mass points. Throughout this solution, we will use $W_P$ to denote the weight of point $P.$ Let $W_C=1.$ Since $3AD=DC,$ it follows that $W_A=3$ and $W_D=W_C+W_A=4.$ Since $AE=EB$ and $W_A=3,$ it follows that $W_B=3.$ Now we focus on $\overline{BD}:$ Since $W_B=3$ and $W_D=4,$ we have $\frac{DF}{FB}=\frac xy=\frac34.$ Therefore, the answer is \[\frac{x}{x+y}=\frac{3}{3+4}=\boxed{\textbf{(A) } \dfrac{3}{7}}.\]

不失一般性，令 $\triangle ABC$ 的边长为 $AB=2, BC=2\sqrt{3},$ 和 $AC=4$。令 $D$ 为从 $B$ 到 $\overline{AC}$ 的垂足，$E$ 为 $\overline{AB}$ 的中点，$F$ 为 $\overline{CE}$ 和 $\overline{BD}$ 的交点。注意 $\triangle ADB$ 和 $\triangle BDC$ 均为 $30^\circ{-}60^\circ{-}90^\circ$ 三角形。由边长比例，得 $AD=1$ 和 $DC=3$。我们得到如下图：此处我们使用质量点法。全篇中用 $W_P$ 表示点 $P$ 的质量。令 $W_C=1$。由于 $3AD=DC$，有 $W_A=3$ 和 $W_D=W_C+W_A=4$。由于 $AE=EB$ 和 $W_A=3$，有 $W_B=3$。现在关注 $\overline{BD}$：由于 $W_B=3$ 和 $W_D=4$，我们有 $\frac{DF}{FB}=\frac xy=\frac34$。因此答案为 \[\frac{x}{x+y}=\frac{3}{3+4}=\boxed{\textbf{(A) } \dfrac{3}{7}}.\]

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