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06 Feb 2020, 03:05
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Difficulty:
95%
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Question Stats:
42% (03:04) correct
58%
(02:42)
wrong
based on 57
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In the diagram given below, the circle and the square have the same center O and equal areas. The circle has radius 1 and intersects one side of the square at P and Q. What is the length of PQ?
A. \(\sqrt{4 − π}\)
B. 1
C. \(\sqrt{π − 1}\)
D. 3/2
E. 2
Are You Up For the Challenge: 700 Level Questions
QA_sectional_01_19.jpg [ 7.66 KiB | Viewed 3539 times ]
A. \(\sqrt{4 − π}\)
B. 1
C. \(\sqrt{π − 1}\)
D. 3/2
E. 2
Are You Up For the Challenge: 700 Level Questions
Attachment:
QA_sectional_01_19.jpg [ 7.66 KiB | Viewed 3539 times ]
06 Feb 2020, 03:33
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Explanation: Equation of circle centered at the origin with radius 1:
x^2+y^2=1
Area of circle: A =πr^2= π
Circle of side length s has the same area:
s^2=π
s=√π
The right side of the square has an equation:
x=√π/2
Plug this into the equation of a circle to find y-coordinates where the line intersects circle:
(√π/2)^2 + y^2 = 1
y^2=1 − π/4
y=± √(4−π)/2
Length of PQ is simply the difference in y-coordinates (since x-coordinates are the same)
|PQ|=(√(4−π)/2)−(−√(4−π)/2)
|PQ|=√(4−π)
IMO-A
x^2+y^2=1
Area of circle: A =πr^2= π
Circle of side length s has the same area:
s^2=π
s=√π
The right side of the square has an equation:
x=√π/2
Plug this into the equation of a circle to find y-coordinates where the line intersects circle:
(√π/2)^2 + y^2 = 1
y^2=1 − π/4
y=± √(4−π)/2
Length of PQ is simply the difference in y-coordinates (since x-coordinates are the same)
|PQ|=(√(4−π)/2)−(−√(4−π)/2)
|PQ|=√(4−π)
IMO-A
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File comment: Figure
main-qimg-a39bc412e500110cc37bba904befdd8f.jpg [ 23.43 KiB | Viewed 3256 times ]
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