I2.1 一個等邊三角形及一個正六邊形的周長比率為 $1: 1$ 。若三角形與六邊形的面積比率為 $2: a$, 求 $a$ 的值。
Let the ratio of perimeter of an equilateral triangle to the perimeter of a regular hexagon be $1: 1$. If the ratio of the area of the triangle to the area of the hexagon is $2: a$, determine the value of $a$.
Let the length of the equilateral triangle be $x$, and that of the regular hexagon be $y$.
Since they have equal perimeter, $3 x=6 y$
$$
\therefore x=2 y
$$

The hexagon can be divided into 6 identical equilateral triangles.
$$
\begin{array}{l}
\text { Ratio of areas }=\frac{1}{2} x^{2} \sin 60^{\circ}: 6 \times \frac{1}{2} y^{2} \sin 60^{\circ}=2: a \\
x^{2}: 6 y^{2}=2: a \\
(2 y)^{2}: 6 y^{2}=2: a \\
\Rightarrow a=3
\end{array}
$$