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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 trianlges is
  • a)
    144 cm
  • b)
    212 cm
  • c)
    288 cm
  • d)
    None of these
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
One side of an equilateral triangle is 24 cm. The mid-points of its si...
Solution:

Given, the side of an equilateral triangle = 24 cm

We need to find the sum of the perimeters of all the triangles formed by joining the mid-points of its sides.

Let's start by drawing the equilateral triangle ABC with side 24 cm.

![image.png](attachment:image.png)

Finding the length of the line segments joining mid-points of the sides:

Let D, E, and F be the mid-points of AB, BC, and CA respectively.

We know that the mid-point of a side of a triangle divides it into two equal parts.

Therefore, BD = AD = CE = BE = AF = CF = 12 cm

Now, we join the mid-points of the sides to form another triangle DEF.

![image-2.png](attachment:image-2.png)

We repeat this process indefinitely to form an infinite number of triangles.

To find the sum of the perimeters of all the triangles formed, we need to find the perimeter of each triangle and add them up.

Perimeter of the first equilateral triangle ABC = 3 × 24 = 72 cm

Perimeter of the second triangle DEF = 3 × 12 = 36 cm

Perimeter of the third triangle GHIJ = 3 × 6 = 18 cm

Perimeter of the fourth triangle KLMNO = 3 × 3 = 9 cm

And so on...

We can see that the perimeter of each successive triangle is half of the previous one.

Therefore, the sum of the perimeters of all the triangles formed is:

72 + 36 + 18 + 9 + ... (infinite series)

This is a geometric series with first term a = 72 and common ratio r = 1/2.

The sum of an infinite geometric series is given by:

S = a/(1-r)

Substituting the values, we get:

S = 72/(1-1/2) = 72/1/2 = 144 cm

Therefore, the sum of the perimeters of all the triangles formed is 144 cm.

Hence, the correct option is (a) 144 cm.
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