One side of an equilateral triangle is 32 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. Find the sum of the perimeters of all the triangles.
Explanation:
We know, in a triangle, line joining the mid-points of two sides is parallel to the third side and half of it.
∴ Sides of the triangle formed by joining midpoints of the sides of a triangle will be half of the original triangle.
Perimeter of the original triangle = 3 × 32 = 96 Perimeter of the triangle formed by joining the mid-points of the previous triangle = 3 × 16 = 48 Perimeter of the triangle formed by joining the mid-points of the previous triangle = 3 × 8 = 24 . . .
∴ Total perimeter = 96 + 48 + 24 + …
= 961-12 = 96 × 2 = 192 cm.
Hence, option (a).
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