One side of an equilateral triangle is 18 cm. The mid-point of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.

Answer

Sum of the perimeters of all the triangles is `108 cm`.

Sum of the Area of all the triangles is `108sqrt(3) cm^2`

Explanation

ATQ,

the midpoints of the triangles are joined successively to get another term and this is being a repeatedly infinite number of terms.

So we will have an infinite number of side length for an infinite number of triangles.



Let us consider `triangleABC` which represents the equilateral triangle with side 18 cm.

D,Eand F are the midpoints of side AB,BC and AC respectively

`:. triangleDEF` represents another equilateral triangle.

Now we can find the length of DE using midpoint theorem of triangle.

If the midpoint of the 2 sides of a triangle are joined, is parallel to the third side and is equal to 21 of it.


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