The sides of a right angled triangle are 6,8 and 10 cm. A new triangle is formed by joining the mid-points of this triangle, again a new triangle is formed by joining the mid points of the new triangle and this process goes on till infinity. Find the total area of such triangle formed.

Options

A. a)`64 cm^2`
B. b)`128cm^2`
C. c)`32cm^2`
D. d)`16cm^2`

Answer

To find the total area of the triangles formed by continuously joining the midpoints of a right-angled triangle with sides 6 cm, 8 cm, and 10 cm, we can follow these steps:

### Step 1: Calculate the area of the original triangle.
The area \( A_1 \) of a right-angled triangle can be calculated using the formula:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

In this case, we can take the base as 6 cm and the height as 8 cm.

\[ A_1 = \frac{1}{2} \times 6 \times 8 = \frac{48}{2} = 24 \text{ square cm} \]

### Step 2: Determine the area of the triangle formed by joining the midpoints.
When we join the midpoints of a triangle, the area of the new triangle \( A_2 \) is \( \frac{1}{4} \) of the area of the original triangle \( A_1 \).

\[ A_2 = \frac{1}{4} A_1 = \frac{1}{4} \times 24 = 6 \text{ square cm} \]

### Step 3: Find the area of subsequent triangles.
Continuing this process, the area of the next triangle \( A_3 \) formed by joining the midpoints of triangle \( A_2 \) will also be \( \frac{1}{4} \) of \( A_2 \).

\[ A_3 = \frac{1}{4} A_2 = \frac{1}{4} \times 6 = 1.5 \text{ square cm} \]

### Step 4: Establish a pattern for the areas.
The areas of the triangles form a geometric series:

- \( A_1 = 24 \)
- \( A_2 = 6 \)
- \( A_3 = 1.5 \)

In general, the area of the \( n \)-th triangle can be expressed as:

\[ A_n = \frac{1}{4^{(n-1)}} \times 24 \]

### Step 5: Calculate the total area of all triangles.
The total area \( S \) of all triangles can be calculated using the formula for the sum of an infinite geometric series:

\[ S = \frac{A}{1 - r} \]

where \( A \) is the first term and \( r \) is the common ratio. Here, \( A = 24 \) and \( r = \frac{1}{4} \).

\[ S = \frac{24}{1 - \frac{1}{4}} = \frac{24}{\frac{3}{4}} = 24 \times \frac{4}{3} = 32 \text{ square cm} \]

### Final Answer:
The total area of all the triangles formed is \( 32 \text{ square cm} \).

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