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Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice?

I want a step by step explanation

 Feb 5, 2021
 #1
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i am not sure about this, but is it just 6?

 

i don't really know, but i can't think of any other possible triangles. i probably missed something...

 

please correct me if i am wrong:)

 Feb 5, 2021
 #2
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Yes, the answer is 6.

 Feb 5, 2021
 #3
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Actually I got it, the answer is 8.

Number the points clockwise, beginning with the upper left as 1. Number the center point 7.

We can create six equilateral triangles with side length one: 176, 172, 273, 657, 574, and 473.

We can also create two equilateral triangles with side length sqrt(3) : 135 and 246.

Thus, there are  8 such equilateral triangles.

vockeyvockvock  Feb 5, 2021
 #4
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ohh, i see, i forgot about the \(\sqrt 3\) ones. 

 

nice job!

macar0ni  Feb 5, 2021

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