Given that the largest triangle is equilateral.....

Question

Given that the largest triangle is equilateral and all the smaller triangles are formed by connecting the midpoints of the sides of the larger triangle that contains, them, how many equilateral triangles are there?

 

Please help me with this

please add the steps to how you got the answer too it would be of much help

Thank you!!!

Answer 1

Hi,

This equilateral triangle will be divided into 4 small equilateral triangles when the midpoints of the large triangle are connected to form the small triangles. If you stop there, then there will be (1 + 4 = 5) five triangles.

 

However, if the problem is to analyze the pattern, the following will explain sum of the triangles formed:

the large triangle + 4 small triangles + 16 smaller triangles + 64 even smaller triangles ...

1 + 4 + 16 + 64 + ...

             Using this pattern you can write the summation as follows:

    2
1. ∑ (4^(n-1)) = 5
  n=1

           

            3

        2. ∑ (4^(n-1)) = 21

          n=1

 

   4
3. ∑ (4^(n-1)) = 85
   n=1

 

     etc.

 

I think this is helpful.

 

SN

Answer 2

Well, here's a quick proof: Each angle in the main triangle is 60 degrees. Since the outer three triangles have the same sides, they are all congruent by SAS. The middle triangle is merely the bases of the outer triangles, which are the same by corresponding parts. So the middle triangle is equilateral. The base angles of the outer three angles must be 180-60=120. Since their outer edges are equal, the base angles must be the same. So 120/2 = 60. So the three angles inside each of the outer triangles are 60 degrees, which makes them equilateral. There's your four equilaterals.