Triangle $A C H$ is determined by three vertices of a cube, see the figure. The height of this triangle to the side $C H$ has a length of $12 \text{~cm}$.

Calculate the area of triangle $A C H$ and the length of the edge of the corresponding cube.

(M. Krejčová)

![](https://cdn.mathpix.com/cropped/2024_04_17_4522893cfc20cd4c648eg-2.jpg?height=554&width=506&top_left_y=474&top_left_x=775)
Each side of the triangle $A C H$ is the diagonal of some face of the cube. The faces of the cube are congruent squares, so the triangle $A C H$ is equilateral.

The relationship between the height $v$ and the side $b$ of an equilateral triangle is $v=\frac{\sqrt{3}}{2} b$. The length of the side of the triangle $A C H$ is therefore

$$
b=\frac{2 \cdot 12}{\sqrt{3}}=8 \sqrt{3} \doteq 13.9(\mathrm{~cm})
$$

The area of a triangle with side length $b$ and corresponding height $v$ is $S=\frac{1}{2} b \cdot v$. The area of the triangle $A C H$ is therefore

$$
S=\frac{12 \cdot 8 \sqrt{3}}{2}=48 \sqrt{3} \doteq 83.1\left(\mathrm{~cm}^{2}\right)
$$

The relationship between the side length $a$ and the diagonal $b$ of a square is $b=a \sqrt{2}$. The length of the edge of the cube is therefore

$$
a=\frac{8 \sqrt{3}}{\sqrt{2}}=4 \sqrt{6} \doteq 9.8(\mathrm{~cm})
$$

Evaluation. 1 point for recognizing the equilateral nature of the triangle $A C H$, expressing $b$, $S$, and $a$; 2 points based on the quality of the commentary.

Notes. The relationships $v=\frac{\sqrt{3}}{2} b$ and $b=a \sqrt{2}$ can be found in tables or derived using the Pythagorean theorem.

The way expressions with square roots are expressed may slightly differ in the results after rounding. Such differences do not affect the evaluation of the problem.