An equilateral triangle is drawn by joining the midpoints of the sides of another equilateral triangle. A third equilateral triangle is drawn inside the second one joining the midpoints of the sides of the second equilateral tringle, and the process continues infinitely. Find the sum of the perimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.

Options

A. 288 units
B. 72 units
C. 36 units
D. 144 units

Answer

To solve the problem, we need to find the sum of the perimeters of all the equilateral triangles formed by joining the midpoints of the sides of the previous triangle, starting with the largest triangle having a side length of 24 units.

### Step 1: Calculate the perimeter of the largest triangle
The perimeter \( P \) of an equilateral triangle is given by the formula:
\[ P = 3 \times \text{side length} \]
For the largest triangle with a side length of 24 units:
\[ P_1 = 3 \times 24 = 72 \text{ units} \]

### Step 2: Determine the side length of the second triangle
The second triangle is formed by joining the midpoints of the sides of the first triangle. The side length of the second triangle will be half of the side length of the first triangle:
\[ \text{Side length of second triangle} = \frac{24}{2} = 12 \text{ units} \]

### Step 3: Calculate the perimeter of the second triangle
Using the same perimeter formula:
\[ P_2 = 3 \times 12 = 36 \text{ units} \]

### Step 4: Determine the side length of the third triangle
Similarly, the side length of the third triangle will be half of the side length of the second triangle:
\[ \text{Side length of third triangle} = \frac{12}{2} = 6 \text{ units} \]

### Step 5: Calculate the perimeter of the third triangle
Using the perimeter formula again:
\[ P_3 = 3 \times 6 = 18 \text{ units} \]

### Step 6: Identify the pattern in the perimeters
We can observe that each subsequent triangle's perimeter is half of the previous triangle's perimeter:
- \( P_1 = 72 \)
- \( P_2 = 36 \)
- \( P_3 = 18 \)

This forms a geometric series where:
- The first term \( a = 72 \)
- The common ratio \( r = \frac{1}{2} \)

### Step 7: Calculate the sum of the infinite geometric series
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{72}{1 - \frac{1}{2}} = \frac{72}{\frac{1}{2}} = 72 \times 2 = 144 \text{ units} \]

### Final Answer
The sum of the perimeters of all the equilateral triangles is \( \boxed{144} \) units.