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08 Jan 2023, 03:37
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D
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Difficulty:
75%
(hard)
Question Stats:
15% (01:27) correct
85%
(01:40)
wrong
based on 27
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Two circles with radii R and r are externally tangent to each other. If line L is tangent to the circles at points M and N, in terms of R and r, what is the value of MN?
A. \(R + r\)
B. \(R*r\)
C. \(\sqrt{R*r}\)
D. \(2\sqrt{R*r}\)
E. \(\sqrt{R^2+r^2}\)
A. \(R + r\)
B. \(R*r\)
C. \(\sqrt{R*r}\)
D. \(2\sqrt{R*r}\)
E. \(\sqrt{R^2+r^2}\)
08 Jan 2023, 08:15
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Bunuel
Let the two circles be as shown.
\(\angle XMN = \angle YNM = 90\)
Join YZ such that \(\angle YZX = 90\)
MZYN is a rectangle
In \(\triangle XZY\)
\(ZX^2 + ZY^2 = XY^2\)
\(ZY^2 = XY^2 - ZX^2 \)
\(ZY^2 = (R+r)^2 - (R-r)^2\)
\(ZY^2 = 4 R*r\)
\(ZY = 2 \sqrt{R*r}\)
\(ZY = MN = 2 \sqrt{R*r}\)
Option D
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