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Mathematics-Online problems:

# Solution to the problem of the (previous) week

Problem:

The figure shows a regular hexagon with side length 1, whose vertices are connected with each other.

Determine the lengths of the line segments and the areas of the triangles and quadrilaterals generated by connecting all the vertices.

 Length of the line segments: Areas:

(The results should be correct to three decimal places.)

Solution:

The rhombus shown above is dissected into two equal parts by the horizontal line. Thus we obtain the two angles shown in the figure. Hence and which leads to and

is the altitude in the equilateral triangle with side length that is Then we have Due to this and we at once obtain Finally, we get and hence

By symmetry, the three areas can also be identified as shown here.
When we calculated the lengths of the line segments, we obtained for the altitude of the equilateral triangle. Hence, its area is Consequently, we have

If, in addition, we draw the third median, we find that by symmetry all the resulting areas are equal and hence Considering also that we obtain

[problem of the week]