Question

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

• A. $144$cm
• B. $212$ cm
• C. $288$ cm
• D. $172$ cm

Answer 1

The correct option is A $144$cm

Side$=24$cm

Side of $2^{nd}$ equilateral triangle$=12$cm

Similarly, side of third triangle$=6$cm

Sides$=3,1.5,\frac{3}{4}\dots ..$

Perimeter of $1^{st}=24+24+24=72cm$

Perimeter of $2^{nd}=12+12+12=36$

Perimeter of $3^{rd}=6+6+6=18$.

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

This is G.P common ratio$=\frac{36}{72}=\frac{1}{2}$

$S_{\infty }=\frac{a}{1-r}=\frac{72}{1-\frac{1}{2}}$

$=144$.