Isosceles triangle `T_1` has a base of 12 meters and a height of 20 meters . The vertices of a second triangle `T_2` are the midpoints of the sides of `T_1`. The vertices of a third triangle , `T_3` , are the midpoints of the sides of `T_2`. Assume the process continues indefinitely , with the vertices of `T_(k+1)` being the midpoints of the sides of `T_k` for every positive integer k. What is the sum of the areas, in square meters, of `T_1,T_2,T_3`, ..... ?

Options

A. 30
B. 40
C. 120
D. 160

Answer

To find the sum of the areas of the isosceles triangles \( T_1, T_2, T_3, \ldots \), we will follow these steps:

### Step 1: Calculate the area of triangle \( T_1 \)
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For triangle \( T_1 \), the base is 12 meters and the height is 20 meters. Thus, we have:
\[ A_{T_1} = \frac{1}{2} \times 12 \times 20 = \frac{240}{2} = 120 \text{ square meters} \]

### Step 2: Determine the area of triangle \( T_2 \)
Triangle \( T_2 \) is formed by connecting the midpoints of the sides of triangle \( T_1 \). The area of triangle \( T_2 \) is one-fourth of the area of triangle \( T_1 \):
\[ A_{T_2} = \frac{1}{4} A_{T_1} = \frac{1}{4} \times 120 = 30 \text{ square meters} \]

### Step 3: Determine the area of triangle \( T_3 \)
Similarly, triangle \( T_3 \) is formed by connecting the midpoints of the sides of triangle \( T_2 \). Thus, the area of triangle \( T_3 \) is:
\[ A_{T_3} = \frac{1}{4} A_{T_2} = \frac{1}{4} \times 30 = 7.5 \text{ square meters} \]

### Step 4: Generalize the area of triangle \( T_k \)
Continuing this process, we can see that the area of triangle \( T_k \) can be expressed as:
\[ A_{T_k} = \frac{1}{4} A_{T_{k-1}} \]
This means that the area of each subsequent triangle is one-fourth of the area of the previous triangle.

### Step 5: Write the series for the total area
The total area \( S \) of all triangles can be expressed as an infinite series:
\[ S = A_{T_1} + A_{T_2} + A_{T_3} + \ldots = 120 + 30 + 7.5 + \ldots \]
This series is a geometric series where:
- The first term \( a = 120 \)
- The common ratio \( r = \frac{1}{4} \)

### Step 6: Calculate the sum of the infinite series
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{120}{1 - \frac{1}{4}} = \frac{120}{\frac{3}{4}} = 120 \times \frac{4}{3} = 160 \text{ square meters} \]

### Final Answer
The sum of the areas of triangles \( T_1, T_2, T_3, \ldots \) is:
\[
\boxed{160} \text{ square meters}