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

Step 1: Understanding the Concept.
Joining the midpoints of a triangle's sides creates a new triangle whose side is exactly half the original side (the midsegment theorem).

Step 2: Find the side lengths of successive triangles.
Triangle 1: side 24 cm. Triangle 2: side 12 cm. Triangle 3: side 6 cm, and so on, each time halving.

Step 3: Find the perimeters.
Perimeter 1 = \(3\times24=72\) cm. Perimeter 2 = \(3\times12=36\) cm. Perimeter 3 = \(3\times6=18\) cm, and so on, forming a geometric progression with first term 72 and common ratio \(\dfrac{1}{2}\).

Step 4: Sum the infinite geometric progression.
Sum \(=\dfrac{a}{1-r}=\dfrac{72}{1-\frac{1}{2}}=\dfrac{72}{\frac{1}{2}}=144\).

Step 5: Final Answer.
The sum of all the perimeters is 144 cm, so option A is correct.