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11 Jun 2020, 04:59
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D
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Difficulty:
65%
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Question Stats:
54% (02:36) correct
46%
(01:45)
wrong
based on 28
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In the figure above four identical circles of radius 19.5 cm have been drawn on the sides of a square such that the points of contact of the circles are the vertices of the square as shown below. What is the perimeter of the square?
A. 92 cm
B. 88 cm
C. 100 cm
D. 110 cm
E. 115 cm
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MayankSingh
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11 Jun 2020, 05:33
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IMO D
Refer attached Image.
Let ABCD be the square & O1, O2, O3, O4 be the center of the circles.
Let join the center of the circles to each other & opposite vertices of the sqaure.
Now, diagonal AC & BD are acting as tangent to the circle at point of contact.
And, Tangent is perpendicular to the radius at point of contact .
So, AC perpendicular to O1O2 & BD AC perpendicular to O2O3
In Quadrilateral AEBO2, Angle AO2B = 90
so AB = Sqrt (R^2+R^2) = 19.5 Sqrt 2
Perimeter = 4x19.5 Sqrt2 = 110
Refer attached Image.
Let ABCD be the square & O1, O2, O3, O4 be the center of the circles.
Let join the center of the circles to each other & opposite vertices of the sqaure.
Now, diagonal AC & BD are acting as tangent to the circle at point of contact.
And, Tangent is perpendicular to the radius at point of contact .
So, AC perpendicular to O1O2 & BD AC perpendicular to O2O3
In Quadrilateral AEBO2, Angle AO2B = 90
so AB = Sqrt (R^2+R^2) = 19.5 Sqrt 2
Perimeter = 4x19.5 Sqrt2 = 110
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WhatsApp Image 2020-06-11 at 5.55.32 PM.jpeg [ 90.89 KiB | Viewed 2913 times ]
yashikaaggarwal
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11 Jun 2020, 05:34
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Draw line from centre to the chord of circles.
We will get a square with side 19.5+19.5 = 39
Since a square edge measure = 90°
The triangle form by radius and chord is a right angle triangle, right angled at Centre O.
The radius are the two legs of right angle triangle and chord also the side of short square = hypotenuse.
Therefore, R^2+R^2 = H^2
2(R)^2 = H^2
2*(19.5)^2 = H^2
380.25*2 = H^2
H = √760.5
H = 27.5 = Side of Square
Perimeter of square = 4*Side
Perimeter = 4*27.5 = 110
Answer is D
Posted from my mobile device
We will get a square with side 19.5+19.5 = 39
Since a square edge measure = 90°
The triangle form by radius and chord is a right angle triangle, right angled at Centre O.
The radius are the two legs of right angle triangle and chord also the side of short square = hypotenuse.
Therefore, R^2+R^2 = H^2
2(R)^2 = H^2
2*(19.5)^2 = H^2
380.25*2 = H^2
H = √760.5
H = 27.5 = Side of Square
Perimeter of square = 4*Side
Perimeter = 4*27.5 = 110
Answer is D
Posted from my mobile device
