Question

One side of an equilateral triangle is 24 cm. The midpoints of its sides are joined to form another triangle whose midpoints are in turn joined to form still another triangle this process continues indefinitely. The sum of the perimeters of all the triangles is

Answer 1

Let a be the side length of equilateral triangle, than `AB=BC=CA=a`
`therefore D,E,F` are the mid-points of BC,CA and AB, respectively.
`therefore EF=FD=DE=(a)/(2)`
and H,I,J are the mid-points of EF, FD and DE, respectively.
`therefore IJ=JH=HI=(a)/(4)`
Similarly, `KL=ML=KM=(a)/(8),"..... "`
P=Sum of perimetes `=3(a+(a)/(2)+(a)/(4)+(a)/(8)+"......")`
`=3((a)/(1-(1)/(2)))=6a=6xx24=144 " cm " " "[ therefore a=24 " cm "]`
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