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2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?
- a)2π : 3
- b)π : 3
- c)√3 : 2π
- d)1 : π
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
2 circles of the same radius are on a 2-D plane that exists such that ...
In the above figure, we need to find the Area of BO'CO
Since B and C are on the circumference of the circle, BO=CO=BO'=CO' =r . Also OO' =r
We can say that BOO' and COO' are 2 equilateral triangles with sides r
Thus area of BO'CO = area BOO' + area COO'
Thus area of BO'CO
Area of 1 circle = πr2
Ratio =
Since B and C are on the circumference of the circle, BO=CO=BO'=CO' =r . Also OO' =r
We can say that BOO' and COO' are 2 equilateral triangles with sides r
Thus area of BO'CO = area BOO' + area COO'
Thus area of BO'CO
Area of 1 circle = πr2
Ratio =
Most Upvoted Answer
2 circles of the same radius are on a 2-D plane that exists such that ...
Let the radius of the circles be $r$.
Since the center of one circle lies on the circumference of the other circle, the distance between the centers of the circles is also $r$.
Let $O_1$ and $O_2$ be the centers of the circles, and let $A$ and $B$ be the points where the circles intersect. Let $C$ be the intersection of $O_1O_2$ and $AB$.
[asy] unitsize(2cm); pair O1,O2,A,B,C; O1 = (0,0); O2 = (1,0); A = intersectionpoints(Circle(O1,1),Circle(O2,1))[0]; B = intersectionpoints(Circle(O1,1),Circle(O2,1))[1]; C = extension(O1,O2,A,B); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(O1--A--B--O2--cycle); dot(O1); dot(O2); dot(A); dot(B); dot(C); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,S); [/asy]
$\triangle O_1O_2C$ is an isosceles triangle with $O_1O_2 = O_1C = O_2C = r$.
Since $\angle O_1O_2C = \angle O_1CO_2 = \angle O_2CO_1 = 90^\circ$, $\triangle O_1O_2C$ is an isosceles right triangle.
Therefore, $\angle O_1OC = \frac{1}{2} \cdot 90^\circ = 45^\circ$.
The area of the sector $AO_1B$ is $\frac{1}{8}$ of the area of the circle centered at $O_1$, because $\angle AOB = \frac{1}{2} \cdot 45^\circ = 22.5^\circ$, so the area of sector $AO_1B$ is $\frac{22.5^\circ}{360^\circ} = \frac{1}{8}$ of the total circle.
The area of triangle $CO_1O_2$ is $\frac{1}{2} \cdot O_1O_2 \cdot O_1C = \frac{1}{2} \cdot r \cdot r = \frac{1}{2}r^2$.
Therefore, the area of quadrilateral $ACOB$ is $\frac{1}{8}$ of the area of the circle plus $\frac{1}{2}r^2$.
The area of the circle is $\pi r^2$.
Therefore, the ratio of the area of quadrilateral $ACOB$ to the area of the circle is $\frac{\frac{1}{8}\pi r^2 + \frac{1}{2}r^2}{\pi r^2} = \boxed{\frac{2}{\pi}+ \frac{1}{4}}$.
Since the center of one circle lies on the circumference of the other circle, the distance between the centers of the circles is also $r$.
Let $O_1$ and $O_2$ be the centers of the circles, and let $A$ and $B$ be the points where the circles intersect. Let $C$ be the intersection of $O_1O_2$ and $AB$.
[asy] unitsize(2cm); pair O1,O2,A,B,C; O1 = (0,0); O2 = (1,0); A = intersectionpoints(Circle(O1,1),Circle(O2,1))[0]; B = intersectionpoints(Circle(O1,1),Circle(O2,1))[1]; C = extension(O1,O2,A,B); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(O1--A--B--O2--cycle); dot(O1); dot(O2); dot(A); dot(B); dot(C); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,S); [/asy]
$\triangle O_1O_2C$ is an isosceles triangle with $O_1O_2 = O_1C = O_2C = r$.
Since $\angle O_1O_2C = \angle O_1CO_2 = \angle O_2CO_1 = 90^\circ$, $\triangle O_1O_2C$ is an isosceles right triangle.
Therefore, $\angle O_1OC = \frac{1}{2} \cdot 90^\circ = 45^\circ$.
The area of the sector $AO_1B$ is $\frac{1}{8}$ of the area of the circle centered at $O_1$, because $\angle AOB = \frac{1}{2} \cdot 45^\circ = 22.5^\circ$, so the area of sector $AO_1B$ is $\frac{22.5^\circ}{360^\circ} = \frac{1}{8}$ of the total circle.
The area of triangle $CO_1O_2$ is $\frac{1}{2} \cdot O_1O_2 \cdot O_1C = \frac{1}{2} \cdot r \cdot r = \frac{1}{2}r^2$.
Therefore, the area of quadrilateral $ACOB$ is $\frac{1}{8}$ of the area of the circle plus $\frac{1}{2}r^2$.
The area of the circle is $\pi r^2$.
Therefore, the ratio of the area of quadrilateral $ACOB$ to the area of the circle is $\frac{\frac{1}{8}\pi r^2 + \frac{1}{2}r^2}{\pi r^2} = \boxed{\frac{2}{\pi}+ \frac{1}{4}}$.
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2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. Can you explain this answer?
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2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. 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 2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. Can you explain this answer?.
2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. 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 2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. Can you explain this answer?.
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2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for 2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of 2 circles of the same radius are on a 2-D plane that exists such that the centre of one circle lies on the circumference of another circle. A quadrilateral is constructed by taking the two points of intersection of the circles and their centres. What is the ratio of the area of the quadrilateral to the area of 1 circle?a)2π : 3b)π : 3c)√3: 2πd)1 :πCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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