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A regular hexagon of side 6 has its center at the origin and one diagonal along the x-axis. Find the coordinates of its vertices.
Solution:
A regular hexagon has 3 longest diagonals that passes thru the center. There are 9 diagonals of a regular hexagon in total. In this problem, let's consider a longest diagonal that lies along the x-axis. To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
There are 6 triangles inside the regular hexagon. Since all sides of a regular hexagon are equal, then it follows that the three longest diagonals are equal to each other and bisect each other at the center. If the two sides of each triangles are equal, then all triangles are isosceles triangles. Let's further analyze and label the figure as follows
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| Photo by Math Principles in Everyday Life |
The vertex angle of each triangles can be calculated as follows
If the two sides of an isosceles triangle are congruent, then it follows that the base angles are congruent also. The base angle of an isosceles triangle is
Since all angles of a triangle are congruent, then all triangles in a regular hexagon are equiangular or equilateral. Let's consider one triangle in a regular hexagon in order to calculate the altitude or height, we have
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| Photo by Math Principles in Everyday Life |
Apply Pythagorean Theorem in order to solve for the altitude or height, we have
Therefore, the coordinates of the vertices of a regular hexagon are
