Find slope of line that are on a hexagon.

Question

The points $(2, 5)$ and $(6, 5)$ are points in the coordinate plane that are vertices of a hexagon with side length $2$. A line through the point $(0, 0)$ divides the hexagon into two parts of equal area. What is the slope of the line?

I have tried graphing it using Desmos and am looking for an algebraic method to solve the problem.

Thanks!

Answer 1

Assuming the hexagon is regular: If all the sides of the hexagon are of length $2$, then the two given points must be diametrically opposite one another. They can't be neighboring vertices, or even separated by only one vertex in between, because such pairs of vertices cannot span a distance of $4$. Only diametrically opposite points span a distance of $4$.

Then the midpoint of the hexagon is the point exactly in between: $(4, 5)$; any line through that point cuts the hexagon in half. A line passing through that point and the origin has slope

$$ m = \frac{5-0}{4-0} = \frac54 $$

Answer 2

Note that

• the distance between the given points is 4 then the points are two opposite vertices
• for simmetry to divide in equal parts the line must pass throught the midpoint (4,5)

thus the line is given by

$$y=\frac54 x$$

Answer 3

Hint:

The problem has one solution only if the hexagon is regular. In this case see the figure. The searched line is the line $HC$.