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An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.
- a)288 units
- b)72 units
- c)36 units
- d)144 units
Correct answer is option 'D'. Can you explain this answer?
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An equilateral triangle is drawn by joining the midpoints of the sides...
The side of the first equilateral triangle being 24 units, the first perimeter is 72 units. The second
perimeter would be half of that and so on.
72, 36, 18 …
perimeter would be half of that and so on.
72, 36, 18 …
Most Upvoted Answer
An equilateral triangle is drawn by joining the midpoints of the sides...
Given:
- The side length of the largest equilateral triangle is 24 units.
To find:
- The sum of the perimeters of all the equilateral triangles.
Solution:
Let's start by finding the side length of the second equilateral triangle.
Step 1:
- The length of a side of the largest equilateral triangle is 24 units.
- The midpoints of the sides of an equilateral triangle divide the sides into segments of equal length, which is half the length of the original side.
- Therefore, the side length of the second equilateral triangle is 24/2 = 12 units.
Step 2:
- Similarly, the side length of the third equilateral triangle is half the side length of the second equilateral triangle.
- So, the side length of the third equilateral triangle is 12/2 = 6 units.
Step 3:
- We can observe that each subsequent equilateral triangle will have half the side length of the previous one.
- So, the side length of the fourth equilateral triangle is 6/2 = 3 units.
- The side length of the fifth equilateral triangle is 3/2 = 1.5 units.
- And so on...
Step 4:
- We can see that the side length of each subsequent equilateral triangle keeps halving, and it will eventually approach zero as we continue the process infinitely.
- Therefore, the sum of the perimeters of all the equilateral triangles can be calculated as the sum of an infinite geometric series.
Step 5:
- The sum of an infinite geometric series can be calculated using the formula: sum = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.
- In this case, the first term (a) is 24 units (side length of the largest equilateral triangle), and the common ratio (r) is 1/2 (since each subsequent equilateral triangle has half the side length of the previous one).
- Plugging in the values, we get: sum = 24 / (1 - 1/2) = 24 / (1/2) = 24 * 2 = 48 units.
Step 6:
- However, we need to consider that the first equilateral triangle (the largest one) is not part of the infinite series.
- So, we subtract the side length of the largest equilateral triangle (24 units) from the sum we calculated in the previous step.
- Final sum = 48 - 24 = 24 units.
Conclusion:
- The sum of the perimeters of all the equilateral triangles is 24 units.
- Therefore, the correct answer is option 'D'.
- The side length of the largest equilateral triangle is 24 units.
To find:
- The sum of the perimeters of all the equilateral triangles.
Solution:
Let's start by finding the side length of the second equilateral triangle.
Step 1:
- The length of a side of the largest equilateral triangle is 24 units.
- The midpoints of the sides of an equilateral triangle divide the sides into segments of equal length, which is half the length of the original side.
- Therefore, the side length of the second equilateral triangle is 24/2 = 12 units.
Step 2:
- Similarly, the side length of the third equilateral triangle is half the side length of the second equilateral triangle.
- So, the side length of the third equilateral triangle is 12/2 = 6 units.
Step 3:
- We can observe that each subsequent equilateral triangle will have half the side length of the previous one.
- So, the side length of the fourth equilateral triangle is 6/2 = 3 units.
- The side length of the fifth equilateral triangle is 3/2 = 1.5 units.
- And so on...
Step 4:
- We can see that the side length of each subsequent equilateral triangle keeps halving, and it will eventually approach zero as we continue the process infinitely.
- Therefore, the sum of the perimeters of all the equilateral triangles can be calculated as the sum of an infinite geometric series.
Step 5:
- The sum of an infinite geometric series can be calculated using the formula: sum = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.
- In this case, the first term (a) is 24 units (side length of the largest equilateral triangle), and the common ratio (r) is 1/2 (since each subsequent equilateral triangle has half the side length of the previous one).
- Plugging in the values, we get: sum = 24 / (1 - 1/2) = 24 / (1/2) = 24 * 2 = 48 units.
Step 6:
- However, we need to consider that the first equilateral triangle (the largest one) is not part of the infinite series.
- So, we subtract the side length of the largest equilateral triangle (24 units) from the sum we calculated in the previous step.
- Final sum = 48 - 24 = 24 units.
Conclusion:
- The sum of the perimeters of all the equilateral triangles is 24 units.
- Therefore, the correct answer is option 'D'.
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An equilateral triangle is drawn by joining the midpoints of the sides...
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An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.a)288 unitsb)72 unitsc)36 unitsd)144 unitsCorrect answer is option 'D'. Can you explain this answer?
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An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.a)288 unitsb)72 unitsc)36 unitsd)144 unitsCorrect answer is option 'D'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.a)288 unitsb)72 unitsc)36 unitsd)144 unitsCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.a)288 unitsb)72 unitsc)36 unitsd)144 unitsCorrect answer is option 'D'. Can you explain this answer?.
An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.a)288 unitsb)72 unitsc)36 unitsd)144 unitsCorrect answer is option 'D'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.a)288 unitsb)72 unitsc)36 unitsd)144 unitsCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.a)288 unitsb)72 unitsc)36 unitsd)144 unitsCorrect answer is option 'D'. Can you explain this answer?.
Solutions for An equilateral triangle is drawn by joining the midpoints of the sides of another equilateraltriangle. A third equilateral triangle is drawn inside the second one joining the midpoints of thesides of the second equilateral triangle, and the process continues infinitely. Find the sum of theperimeters of all the equilateral triangles, if the side of the largest equilateral triangle is 24 units.a)288 unitsb)72 unitsc)36 unitsd)144 unitsCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Quant.
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