Question

One side of an equilateral triangle is 24 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.

• A. 144 cm
• B. 72 cm
• C. 536 cm
• D. 676 cm

Hint:

Each new triangle's perimeter is half the previous one; sum the infinite GP with first term 72 and ratio 1/2.

Answer 1

The Correct Option is A

Since each new triangle's perimeter is exactly half of the one before it, the whole sequence of perimeters is a simple halving pattern that can be summed using the infinite geometric series formula.

1. First perimeter: $3 \times 24 = 72$ cm.
2. Each subsequent triangle's side is half the previous one (midsegment theorem), so each subsequent perimeter is also exactly half the previous perimeter.
3. This gives a geometric series: $72 + 36 + 18 + 9 + \dots$, with common ratio $r = \frac{1}{2}$.
4. Sum to infinity $= \frac{72}{1 - \frac{1}{2}} = \frac{72}{0.5} = 144$.

So the correct answer is option A, 144 cm.