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An isosceles triangle with one angle of 120° is inscribed in a circle
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06 Aug 2021, 21:22
06 Aug 2021, 21:22
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SUPER HARD
An isosceles triangle with one angle of 120° is inscribed in a circle of radius 2 units. The triangle is rotated 90° about the center of the circle.
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
An isosceles triangle with one angle of 120° is inscribed in a circle of radius 2 units. The triangle is rotated 90° about the center of the circle.
Quantity A |
Quantity B |
Total area covered by the triangle throughout this movement, from starting point to final resting point |
5 |
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
Retired Moderator
Joined: 19 Nov 2020
Posts: 326
Given Kudos: 64
GRE 1: Q160 V152
An isosceles triangle with one angle of 120° is inscribed in a circle
[#permalink]
08 Aug 2021, 09:37
08 Aug 2021, 09:37
1
BD is bisector of angle ABC, and BD is diameter. Hence, BD=4
angle ABD=60 and ABD is the right triangle with angles 30-60-90. Hence, AB=2
Thus, AB=BC=radius=2 (this is very important to visualize and understand)
There are, in total, 6 rotations of the side equal to any circle radius, if the side intercepted by this circle is equal to the circle's radius. Only 6 rotations describe the total circle area too.
The side AB is rotated into a new position to become the side ab less than three times. Namely, there are (90+60)/360 or 5/12 times the side AB has been rotated along the circle.
The area of cirle is \(3.14*r^2=3.14*4=12.56\). The part corresponding with how many times the side AB has been rotated is \(12.56*5/12\) of the area for this circle.
The answer rests with the difference between half of the area of ABC and equilateral triangle describing the circle area (explained above). To find the actual area of ABC, we use 30-60-90 right triangle area property again, as angle ABC is bisected into two angles pf 60 degrees. AB is the hypotenuse the height (h) is 1. Further, we don't even have to find the base and compare area of ABC with equilateral triangle (all sides=r). It's enough to check that h=1/2 r and hald of the area of ABC is half of the area of equilateral triangle. Thus, the value obtained earlier is divided by 2 ---> \((12.56*5/12) : 2\) leaving less than 5.
Hence, answer is B
angle ABD=60 and ABD is the right triangle with angles 30-60-90. Hence, AB=2
Thus, AB=BC=radius=2 (this is very important to visualize and understand)
There are, in total, 6 rotations of the side equal to any circle radius, if the side intercepted by this circle is equal to the circle's radius. Only 6 rotations describe the total circle area too.
The side AB is rotated into a new position to become the side ab less than three times. Namely, there are (90+60)/360 or 5/12 times the side AB has been rotated along the circle.
The area of cirle is \(3.14*r^2=3.14*4=12.56\). The part corresponding with how many times the side AB has been rotated is \(12.56*5/12\) of the area for this circle.
The answer rests with the difference between half of the area of ABC and equilateral triangle describing the circle area (explained above). To find the actual area of ABC, we use 30-60-90 right triangle area property again, as angle ABC is bisected into two angles pf 60 degrees. AB is the hypotenuse the height (h) is 1. Further, we don't even have to find the base and compare area of ABC with equilateral triangle (all sides=r). It's enough to check that h=1/2 r and hald of the area of ABC is half of the area of equilateral triangle. Thus, the value obtained earlier is divided by 2 ---> \((12.56*5/12) : 2\) leaving less than 5.
Hence, answer is B
KarunMendiratta wrote:
SUPER HARD
An isosceles triangle with one angle of 120° is inscribed in a circle of radius 2 units. The triangle is rotated 90° about the center of the circle.
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
An isosceles triangle with one angle of 120° is inscribed in a circle of radius 2 units. The triangle is rotated 90° about the center of the circle.
Quantity A |
Quantity B |
Total area covered by the triangle throughout this movement, from starting point to final resting point |
5 |
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
Attachments
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gmatclubot
An isosceles triangle with one angle of 120° is inscribed in a circle [#permalink]
08 Aug 2021, 09:37
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