AMC12 2014 A

AMC12 2014 A · Q10

AMC12 2014 A · Q10. It mainly tests Triangles (properties), Area & perimeter.

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?

在一个边长为1的正三角形的三边上，各构造一个全等等腰三角形。这三个等腰三角形的面积之和等于正三角形的面积。其中一个等腰三角形的两条全等边的长度是多少？

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

(B) \dfrac{\sqrt3}3 \dfrac{\sqrt3}3

(D) \dfrac{\sqrt2}2 \dfrac{\sqrt2}2

(E) \dfrac{\sqrt3}2 \dfrac{\sqrt3}2

Answer

Correct choice: (B)

正确答案：（B）

Solution

Reflect each of the triangles over its respective side. Then since the areas of the triangles total to the area of the equilateral triangle, it can be seen that the triangles fill up the equilateral one and the vertices of these triangles concur at the circumcenter of the equilateral triangle. Hence the desired answer is just its circumradius, or $\boxed{\dfrac{\sqrt3}3\textbf{ (B)}}$.

将每个三角形沿其底边反射。由于面积总和等于正三角形面积，可见这些三角形填满正三角形，且顶点在正三角形的圆心重合。故所求即其外接圆半径，即$\boxed{\dfrac{\sqrt3}3\textbf{ (B)}}$。

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