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# Triangles

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Hi friends,

Last one for the day please....

Triangle ABC has equal sides of 6p. The vertexes of a second triangle inside the first triangle, meets on the midpoints of the sides of the first triangle. The vertexes of a third triangle, inside the second one, meets on the midpoints of the second triangle. Determine the sides of all three triangles in terms of p, for example ABC:DEF:GHK

Also determine the height of the triangles in terms of p.

I said the length of the sides of the 1st triangle is 6p....given

Since the vertexes of the second triangle lies on the midpoints of the first one, they have to be 3p, each...and the same argument for the third one..which must be 1,5p....however, I'm sure there's way much more to this...please help....

Feb 16, 2022

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See here, JM Let p = 1

Note that triangles DEC and ABC are similar

And triangle DEF is just a reflection of triangle DEC

And the scale of   DEC : ABC  =  1/2

So.....the scale of DEF : ABC   =  1/2

i.e......each successive triangle  will be similar  with a scale factor of 1/2 of its predecessor

Each successive triangle  will  have  sides of 6p , 3p , 1.5p    and so on     (each will have 1/2 the side length of its predecessor )

And the succesive  heights follow the same pattern   (3sqrt 3)p , ( 1.5 sqrt 3) p , ( .75 sqrt 3)p  .....etc.   Feb 16, 2022
#2
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Hi CPhill,

Thank you kindly...I was just wondering how would I render the answer mathematically...meaning, how would the sum, answering the question, look like?...I hope my question makes sense?

juriemagic  Feb 16, 2022