Proving the Rationality of the Distance Between Opposite Edges of a Regular Hexagon with Side Length √3

In this article, we will delve into the fascinating world of geometry and explore how to prove that the distance between opposite edges of a regular hexagon with side length √3 is a rational value. We'll cover the steps to visualize the hexagon, find the equations of its edges, and apply the formula for the distance between two parallel lines to compute the distance. Let's get started with the detailed explanation.

Visualizing the Hexagon

A regular hexagon can be inscribed in a circle, with its vertices evenly spaced around the circumference. We can place the hexagon's center at the origin of a coordinate system. For simplicity, the vertices of the regular hexagon with side length √3 can be determined using polar coordinates. Here are the coordinates of the vertices:

A (√3, 0) B (√3/2, 3/2) C (-√3/2, 3/2) D (-√3, 0) E (-√3/2, -3/2) F (√3/2, -3/2)

Finding the Equations of the Edges

The hexagon consists of 6 edges. We can identify pairs of opposite edges as:

AB with DE BC with EF

For the edges AB and DE, their equations can be derived by determining the line passing through the points. The line AB has a negative slope, while the line DE has a positive slope. Let's find the equations of these lines.

The line AB can be represented as:

y -√3x 3√3/2

Rewriting this into the standard line form:

√3x y - 3√3/2 0

Here, C_1 -3√3/2

The line DE can be represented as:

y √3x - 3

Rewriting this into the standard line form:

√3x - y - 3 0

Here, C_2 -3

Calculating the Perpendicular Distance

The distance d between two parallel lines of the form Ax By C_1 0 and Ax By C_2 0 can be calculated using the formula:

d |C_2 - C_1| / √(A^2 B^2)

For the lines AB and DE, we have:

A √3 B 1 C_1 -3√3/2 C_2 -3

Substituting these into the distance formula, we get:

d |-3 - (-3√3/2)| / √((√3)^2 1^2)

Simplifying the numerator and the denominator:

d |3 3√3/2| / √(3 1)

d |3 3√3/2| / 2

d 3{2} 3√3/2}{2}

d 3{2} 3√3{4}

The distance between opposite edges of the regular hexagon with side length √3 is a rational value. Hence, we have:

Distance between opposite edges 3{2} 3√3{4}

Conclusion

This calculation demonstrates that the distance between the opposite edges of a regular hexagon with side length √3 is a rational value. Thus, we can confidently conclude:

Distance between opposite edges 3{2} 3√3{4}