The middle points of the sides of a triangle are joined forming a second triangle. Again a third triangle is formed by joining the middle points of this second triangle and this process is repeated infinitely. If the perimeter and area of the outer triangle are P and A respectively, what will be the sum of perimeters of triangles thus formed?

Options

A. `2p`
B. `p^2`
C. `3^p`
D. `p^2 // 2`

Answer

To solve the problem, we need to analyze the formation of triangles and their perimeters step by step.

### Step 1: Understand the Formation of Triangles
When we join the midpoints of the sides of a triangle, we create a new triangle. This new triangle is similar to the original triangle and its sides are half the length of the corresponding sides of the original triangle.

### Step 2: Determine the Perimeter of Each Triangle
Let the perimeter of the outer triangle be \( P \). The perimeter of the first inner triangle (formed by joining the midpoints of the outer triangle) will be:
\[ \text{Perimeter of first inner triangle} = \frac{P}{2} \]
The second inner triangle will have a perimeter of:
\[ \text{Perimeter of second inner triangle} = \frac{P}{4} \]
Continuing this pattern, the perimeter of the \( n \)-th inner triangle will be:
\[ \text{Perimeter of } n\text{-th inner triangle} = \frac{P}{2^n} \]

### Step 3: Sum the Perimeters of All Triangles
To find the total sum of the perimeters of all triangles formed, we can express this as:
\[ \text{Total Perimeter} = P + \frac{P}{2} + \frac{P}{4} + \frac{P}{8} + \ldots \]
This is an infinite geometric series where the first term \( a = P \) and the common ratio \( r = \frac{1}{2} \).

### Step 4: Use the Formula for the Sum of an 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:
\[ S = \frac{P}{1 - \frac{1}{2}} = \frac{P}{\frac{1}{2}} = 2P \]

### Conclusion
Thus, the total sum of the perimeters of all triangles formed is:
\[ \text{Total Sum of Perimeters} = 2P \]