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

The side of triangle 1 : 24 cm
The side of triangle formed by joining midpoints : 12 cm
and so onn..
Sum of perimeter - 3(24) + 3(24)/2 + 3(24)/4 + 3(24)/8 .....
= 3(24) [ 1 + 1/2 + 1/4 + 1/8....]

The larger triangle will have the perimeter 72 cm. After joining the mid points of the triangle a small triangle also called pedal triangle is formed whose sides are 12 cm and perimeter 36 cm. This process continues to infinite so we have----> Perimeter= 72+36+18+9+............ which forms an infinite Geometric progression with common ratio 1/2. Sum= First Term/(1- common ratio) = 72/(1-0.5) = 144cm

144

the first triangle we form length of the side become half &it continues so long as we calculate 24(3)+12(3)+6(3)+3(3)+3/2(3)+3/4(3)+..........=144

72+72/2+72/4......
72(1/1-1/2)=144