Question

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

Answer 1

The mid-points of its sides are joined to from another triangle whose mid-points are joined to from another triangle. This process continues indefinitely.

Side of 2nd equilateral triangle $=12cm$ ( line segment joining mid-point of two sides of a triangle is parallel to third side and half of it ).

Similarly, side of third equilateral triangle $=6cm....$ sides are $3,1,5,\frac{3}{4}.$

Perimeter of 1st triangle $=24=24+24=72cm$

Perimeter of 2nd triangle $=12+12+12=36cm$

Perimeter of 3rd triangle $=6+6+6=18cm.$

Sum of perimeter of all triangles $=72+36+18+9+\frac{9}{2}+\frac{9}{4}$

This is a GP series where,

common ratio $=\frac{36}{72}=\frac{1}{2}$

Sum of an infinity GP $=\frac{a}{1-r}$

$=\frac{72}{1-\frac{1}{2}}$

$=72\times 2$

$=144.$

Hence, the answer is $144.$