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Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will be
- a)188√3
- b)248√3
- c)164√3
- d)192√3
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Given an equilateral triangle T1 with side 24 cm, a second triangle T2...
We can see that T2 is formed by using the mid points of T1. Hence, we can say that area of triangle of T2 will be (1/4)thof the area of triangle T1.
Area of triangle T1 =√3 / 4 * (24)2 = 144√3 sq.cm
Area of triangle T2 = 144√3 / 4= 36√3 sq. cm
Sum of the area of all triangles = T1 + T2 + T3 + ...
=>T1 + T1 /4 + T1 /42 + ...
=> T1 / 1 – 0.25
=>4 / 3 * T1
=>4 / 3 * 144√3
=>192 √3
Hence, option D is the correct answer.
Most Upvoted Answer
Given an equilateral triangle T1 with side 24 cm, a second triangle T2...
Approach:
To solve this problem, we need to understand that the process described leads to the formation of a series of smaller and smaller equilateral triangles. We can use the concept of geometric progression to find the sum of the areas of infinitely many such triangles.
Solution:
Step 1: Find the area of the original equilateral triangle T1
The area of an equilateral triangle can be calculated using the formula: Area = (√3/4) * side^2.
In this case, the side of T1 is 24 cm, so the area of T1 = (√3/4) * 24^2 = 144√3 sq cm.
Step 2: Find the ratio of areas between consecutive triangles
When we form T2 by joining the midpoints of T1, each side of T2 is half the length of T1. Therefore, the ratio of areas between T1 and T2 is 1:4.
Similarly, the ratio of areas between T2 and T3 is also 1:4.
Step 3: Find the sum of the infinite series of areas
Since the common ratio between consecutive areas is 1:4, we can set up the series as follows:
Sum = Area of T1 + Area of T2 + Area of T3 + ...
Sum = 144√3 + (1/4)*144√3 + (1/16)*144√3 + ...
Sum = 144√3 * (1 + 1/4 + 1/16 + ...)
Sum = 144√3 * (1/(1-1/4))
Sum = 192√3 sq cm
Therefore, the sum of the areas of infinitely many such triangles T1, T2, T3,... will be 192√3 sq cm, which corresponds to option (d).
To solve this problem, we need to understand that the process described leads to the formation of a series of smaller and smaller equilateral triangles. We can use the concept of geometric progression to find the sum of the areas of infinitely many such triangles.
Solution:
Step 1: Find the area of the original equilateral triangle T1
The area of an equilateral triangle can be calculated using the formula: Area = (√3/4) * side^2.
In this case, the side of T1 is 24 cm, so the area of T1 = (√3/4) * 24^2 = 144√3 sq cm.
Step 2: Find the ratio of areas between consecutive triangles
When we form T2 by joining the midpoints of T1, each side of T2 is half the length of T1. Therefore, the ratio of areas between T1 and T2 is 1:4.
Similarly, the ratio of areas between T2 and T3 is also 1:4.
Step 3: Find the sum of the infinite series of areas
Since the common ratio between consecutive areas is 1:4, we can set up the series as follows:
Sum = Area of T1 + Area of T2 + Area of T3 + ...
Sum = 144√3 + (1/4)*144√3 + (1/16)*144√3 + ...
Sum = 144√3 * (1 + 1/4 + 1/16 + ...)
Sum = 144√3 * (1/(1-1/4))
Sum = 192√3 sq cm
Therefore, the sum of the areas of infinitely many such triangles T1, T2, T3,... will be 192√3 sq cm, which corresponds to option (d).
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Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will bea)188√3b)248√3c)164√3d)192√3Correct answer is option 'D'. Can you explain this answer?
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