To find the sum of the perimeters of all the triangles formed by continuously joining the midpoints of an equilateral triangle, we can follow a systematic approach.
Initial Triangle Perimeter
The perimeter of the original equilateral triangle can be calculated as:
- Perimeter = 3 × side length
- Perimeter = 3 × 24 cm = 72 cm
Subsequent Triangles
Each time we form a new triangle by joining the midpoints, the side length of the new triangle is half of the previous triangle's side length. Therefore:
- First triangle: 24 cm (side length)
- Second triangle: 12 cm (side length)
- Third triangle: 6 cm (side length)
- And so on...
Perimeter of Subsequent Triangles
The perimeter of each subsequent triangle can be expressed as:
- Second triangle perimeter = 3 × 12 cm = 36 cm
- Third triangle perimeter = 3 × 6 cm = 18 cm
- Fourth triangle perimeter = 3 × 3 cm = 9 cm
Geometric Series
The perimeters form a geometric series where:
- First term (a) = 72 cm
- Common ratio (r) = 1/2
The sum of an infinite geometric series can be calculated using the formula:
Sum = a / (1 - r)
Substituting the values:
Sum = 72 cm / (1 - 1/2) = 72 cm / (1/2) = 144 cm
Final Result
The total sum of the perimeters of all the triangles formed is 144 cm.
