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....
See here, JM
Your intuition was correct
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.