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 joined to form another triangle. This process continues indefinitely. Find the sum of the perimeter of all the triangles.

Answer 1

Side of an equilateral triangle= 24 cm

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

Side of 2nd equilateral triangle=12 cm [ 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=6 cm

......sides are 3, 1.5,3/4,....

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

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

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

..........

sum of the perimeter of all the triangles=72+36+18+9+9/2+9/4+........

            This is a geometric progression having common ratio =36/72=1/2

Sum of an infinite G.P= $\frac{a}{1-r}$

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

                                = 72 × 2

                                 =144 which is the sum of perimeter of all triangles.

Answer 2

Answer:

The mid-points of its sides are joined to form another triangle whose mid-points are joined to form another triangle. ... =144 which is the sum of perimeter of all triangle