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26 Aug 2021, 08:00
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B
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Difficulty:
45%
(medium)
Question Stats:
65% (01:46) correct
35%
(01:58)
wrong
based on 188
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Three blue circles each having the diameter of d, are tied together with a red belt as shown above. What is the length of the belt?
A. \(\pi d+ d\)
B. \(\pi d+ 3d\)
C. \(2\pi d+ d\)
D. \(3\pi d+ d\)
E. \(3\pi d+ 3d\)
M37-22
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M37-22
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22 Nov 2021, 03:32
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Official Solution:
Three blue circles each having the diameter of \(d\), are tied together with a red belt as shown above. What is the length of the belt?
A. \(\pi d+ d\)
B. \(\pi d+ 3d\)
C. \(2\pi d+ d\)
D. \(3\pi d+ d\)
E. \(3\pi d+ 3d\)
Joint the centers of the circle. We get an equilateral triangle with sides of \(r+r=d\).
Join the centers of the circles and the tangency points (recall that the radius of a circle is perpendicular to the tangent line). We get three rectangles.
The length of the belt is \(d+d+d\) plus three times 120° arc, so one full circumference.
The length of the belt \(=3d+\pi d\)
Answer: B
Three blue circles each having the diameter of \(d\), are tied together with a red belt as shown above. What is the length of the belt?
A. \(\pi d+ d\)
B. \(\pi d+ 3d\)
C. \(2\pi d+ d\)
D. \(3\pi d+ d\)
E. \(3\pi d+ 3d\)
Joint the centers of the circle. We get an equilateral triangle with sides of \(r+r=d\).
Join the centers of the circles and the tangency points (recall that the radius of a circle is perpendicular to the tangent line). We get three rectangles.
The length of the belt is \(d+d+d\) plus three times 120° arc, so one full circumference.
The length of the belt \(=3d+\pi d\)
Answer: B
General Discussion
26 Aug 2021, 08:39
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Bunuel
Construct the diagram as shown below;
As we join the centers of the 3 circles then we get an eqi. triangle with side as d;
When we drop perpendiculars onto the tangents they will become rectangles as the angle made by tangent and center is 90'; (Same radius)
So now the angle made by each arc is 120';
So we know the circumference of one arc is \(\pi\)D/3, But we have 3 arcs so the total circumference is \(\pi\)D.
Belt length = 3D + \(\pi\)D.
IMO B
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