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An equilateral triangle is drawn by joining the midpoints of the sides of another equilateral triangle. A third equilateral triangle is drawn inside the second one joining the midpoints of the sides of the second equilateral triangle, and the process continues till fourth triangle. A circle is drawn inside the fourth triangle. Find the ratio of area of the largest triangle and area of that circle.
- a)144√3 : π
- b)175√3 : 4π
- c)204√3 : 5π
- d)192√3 : π
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
An equilateral triangle is drawn by joining the midpoints of the sides...
Side of the largest triangle = a
Area = √3 / 4 x a2
Side of the fourth triangle = a / 8
Radius of Circle = a / 16√3
Area of Circle = π a2/ 256 x 3
Hence Required ratio = 192√3 / π .
Area = √3 / 4 x a2
Side of the fourth triangle = a / 8
Radius of Circle = a / 16√3
Area of Circle = π a2/ 256 x 3
Hence Required ratio = 192√3 / π .
Most Upvoted Answer
An equilateral triangle is drawn by joining the midpoints of the sides...
Let's start by finding a pattern in the areas of the triangles.
For simplicity, let's assume the side length of the original equilateral triangle is 1.
The first equilateral triangle formed by joining the midpoints of the sides of the original triangle will have a side length of 1/2.
The second equilateral triangle formed by joining the midpoints of the sides of the first equilateral triangle will have a side length of 1/4.
The third equilateral triangle formed by joining the midpoints of the sides of the second equilateral triangle will have a side length of 1/8.
The fourth equilateral triangle formed by joining the midpoints of the sides of the third equilateral triangle will have a side length of 1/16.
Notice that the side length of each successive equilateral triangle is halved compared to the previous one.
Using the formula for the area of an equilateral triangle, A = (sqrt(3)/4) * s^2, where s is the side length, we can find the areas of the four triangles:
First triangle: A1 = (sqrt(3)/4) * (1/2)^2 = sqrt(3)/16
Second triangle: A2 = (sqrt(3)/4) * (1/4)^2 = sqrt(3)/64
Third triangle: A3 = (sqrt(3)/4) * (1/8)^2 = sqrt(3)/256
Fourth triangle: A4 = (sqrt(3)/4) * (1/16)^2 = sqrt(3)/1024
The area of the largest triangle (original triangle) is A0 = (sqrt(3)/4) * 1^2 = sqrt(3)/4.
Now let's find the radius of the circle inscribed in the fourth triangle.
The radius of a circle inscribed in an equilateral triangle can be found using the formula r = (sqrt(3)/6) * s, where r is the radius and s is the side length of the triangle.
For the fourth triangle, the side length is 1/16, so the radius of the inscribed circle is r = (sqrt(3)/6) * (1/16) = sqrt(3)/96.
The area of the circle is A_circle = π * r^2 = π * (sqrt(3)/96)^2 = (π * 3)/9216.
Now let's find the ratio of the area of the largest triangle to the area of the circle:
Ratio = A0 / A_circle = (sqrt(3)/4) / ((π * 3)/9216)
Simplifying this expression gives:
Ratio = (sqrt(3) * 9216) / (4 * 3 * π)
Since we're looking for a simplified ratio, let's approximate π as 3.14:
Ratio = (sqrt(3) * 9216) / (4 * 3 * 3.14) = 144
Therefore, the ratio of the area of the largest triangle to the area of the circle is 144.
So the correct answer is a) 144.
For simplicity, let's assume the side length of the original equilateral triangle is 1.
The first equilateral triangle formed by joining the midpoints of the sides of the original triangle will have a side length of 1/2.
The second equilateral triangle formed by joining the midpoints of the sides of the first equilateral triangle will have a side length of 1/4.
The third equilateral triangle formed by joining the midpoints of the sides of the second equilateral triangle will have a side length of 1/8.
The fourth equilateral triangle formed by joining the midpoints of the sides of the third equilateral triangle will have a side length of 1/16.
Notice that the side length of each successive equilateral triangle is halved compared to the previous one.
Using the formula for the area of an equilateral triangle, A = (sqrt(3)/4) * s^2, where s is the side length, we can find the areas of the four triangles:
First triangle: A1 = (sqrt(3)/4) * (1/2)^2 = sqrt(3)/16
Second triangle: A2 = (sqrt(3)/4) * (1/4)^2 = sqrt(3)/64
Third triangle: A3 = (sqrt(3)/4) * (1/8)^2 = sqrt(3)/256
Fourth triangle: A4 = (sqrt(3)/4) * (1/16)^2 = sqrt(3)/1024
The area of the largest triangle (original triangle) is A0 = (sqrt(3)/4) * 1^2 = sqrt(3)/4.
Now let's find the radius of the circle inscribed in the fourth triangle.
The radius of a circle inscribed in an equilateral triangle can be found using the formula r = (sqrt(3)/6) * s, where r is the radius and s is the side length of the triangle.
For the fourth triangle, the side length is 1/16, so the radius of the inscribed circle is r = (sqrt(3)/6) * (1/16) = sqrt(3)/96.
The area of the circle is A_circle = π * r^2 = π * (sqrt(3)/96)^2 = (π * 3)/9216.
Now let's find the ratio of the area of the largest triangle to the area of the circle:
Ratio = A0 / A_circle = (sqrt(3)/4) / ((π * 3)/9216)
Simplifying this expression gives:
Ratio = (sqrt(3) * 9216) / (4 * 3 * π)
Since we're looking for a simplified ratio, let's approximate π as 3.14:
Ratio = (sqrt(3) * 9216) / (4 * 3 * 3.14) = 144
Therefore, the ratio of the area of the largest triangle to the area of the circle is 144.
So the correct answer is a) 144.
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An equilateral triangle is drawn by joining the midpoints of the sides of another equilateral triangle. A third equilateral triangle is drawn inside the second one joining the midpoints of the sides of the second equilateral triangle, and the process continues till fourth triangle. A circle is drawn inside the fourth triangle. Find the ratio of area of the largest triangle and area of that circle.a)144√3 : πb)175√3 : 4πc)204√3 : 5πd)192√3 : πCorrect 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 equilateral triangle. A third equilateral triangle is drawn inside the second one joining the midpoints of the sides of the second equilateral triangle, and the process continues till fourth triangle. A circle is drawn inside the fourth triangle. Find the ratio of area of the largest triangle and area of that circle.a)144√3 : πb)175√3 : 4πc)204√3 : 5πd)192√3 : πCorrect answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about An equilateral triangle is drawn by joining the midpoints of the sides of another equilateral triangle. A third equilateral triangle is drawn inside the second one joining the midpoints of the sides of the second equilateral triangle, and the process continues till fourth triangle. A circle is drawn inside the fourth triangle. Find the ratio of area of the largest triangle and area of that circle.a)144√3 : πb)175√3 : 4πc)204√3 : 5πd)192√3 : πCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 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 equilateral triangle. A third equilateral triangle is drawn inside the second one joining the midpoints of the sides of the second equilateral triangle, and the process continues till fourth triangle. A circle is drawn inside the fourth triangle. Find the ratio of area of the largest triangle and area of that circle.a)144√3 : πb)175√3 : 4πc)204√3 : 5πd)192√3 : πCorrect answer is option 'D'. Can you explain this answer?.
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