regular hexagon problem: what are the dimensions of those rectangles?

[Night14]

Hey, I've been working on this problem (If you put a square on each side of an equilateral triangle and connect the outside vertices, you get a hexagon. However, this hexagon isn't regular. A regular hexagon can be made if you use rectangles with the correct dimensions, instead of squares. What are the dimensions of those rectangles of the sides of the equilateral triangle have length 1?) for quite a while and haven't been able to solve it. I don't have a clear idea of what I need to do so it would greatly appreciated if someone could explain to me how to solve this if they know what to do. Thanks (Note: This is due tomorrow so I don't have much time ).

 

[stapel]

Hey, I've been working on this problem:

If you put a square on each side of an equilateral triangle and connect the outside vertices, you get a hexagon. However, this hexagon isn't regular. A regular hexagon can be made if you use rectangles with the correct dimensions, instead of squares. What are the dimensions of those rectangles of the sides of the equilateral triangle have length 1?

I've been working on this for quite a while and haven't been able to solve it. I don't have a clear idea of what I need to do so it would greatly appreciated if someone could explain to me how to solve this if they know what to do.

Um... If you "don't have a clear idea of what [you] need to do", then what, exactly, have you been doing "for quite a while"...? :shock:

You started by doing the drawing they talk about, confirming the result they state. You then drew some equilateral triangles with variously sized rectangles, seeing what results you got and trying to determine any patterns. And... then what?

Please be complete. Thank you!

 

[pka]

Hey, I've been working on this problem (If you put a square on each side of an equilateral triangle and connect the outside vertices, you get a hexagon. However, this hexagon isn't regular. A regular hexagon can be made if you use rectangles with the correct dimensions, instead of squares. What are the dimensions of those rectangles of the sides of the equilateral triangle have length 1.

In any equilateral triangle each of the incenter, circumcenter, & orthocenter is the same point.
It is one-third the length of an altitude of the triangle. The "center" of your hexagon is that point. The length of the side of the required hexagon is equal to the length of the equilateral triangle. Thus the other side of the constructing rectangle is two-thirds the distance from the center of the hexagon to one of its sides.

 

[Night14]

So, I started by drawing a diagram and then filling out given information and information that can be concluded. Unknown sides are labeled with the variable x. For the 3 rectangles of the hexagon, the area of each one is 1x. The area of the equilateral triangle in the center is the square root of 3 over 4 using the equilateral triangle area formula. The last 3 triangles have sides 1, x, and x, so I used Heron's formula to find the area of each triangle which ends up being 4x^2-1 all over 16 for each one. I added all of these into an equation and equaled it to the square root of 27 over 2 because the area of the hexagon is that since each side of the hexagon can be concluded as 1 and using the hexagon area formula. So my problem is I'm struggling to solve this equation. I'm pretty sure it's the correct equation but I'm not sure. Here's a picture to help better show my work (the areas of each part of the hexagon are circled).

 

[Night14]

In any equilateral triangle each of the incenter, circumcenter, & orthocenter is the same point.
It is one-third the length of an altitude of the triangle. The "center" of your hexagon is that point. The length of the side of the required hexagon is equal to the length of the equilateral triangle. Thus the other side of the constructing rectangle is two-thirds the distance from the center of the hexagon to one of its sides.

Could you explain this a bit more because I can't follow this logic