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02 Sep 2022, 01:30
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
61% (02:48) correct
39%
(03:20)
wrong
based on 33
sessions
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Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points A and B, as shown in the diagram. The distance AB can be written in the form \(\frac{m}{n}\), where m and n are relatively prime positive integers. What is m + n?
A. 21
B. 29
C. 58
D. 69
E. 93
Attachment:
c1d7b33c122356459463ccf321031d04f8f53752.png [ 7.4 KiB | Viewed 3247 times ]
Are You Up For the Challenge: 700 Level Questions
02 Sep 2022, 03:31
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Let the centre of the largest circle = \(O\)
Let the the line created by the two diameters of the two smaller circles = \(xy\), with the two corresponding points being \(x\) and \(y\) respectively.
We know that \(OA = OB = 13\) and that \(xy = 10\). We can work out the length \(Ox\) and \(Oy\) by subtracting the length the radius of the smaller circles from 13. \(13-5 = 8\). Therefore \(Ox\)=\(Oy\)=\(8\).
\(\triangle OAB\) and \(\triangle Oxy\) are similar and therefore we can go: \(\frac{Ox}{xy} = \frac{OA}{OB}\)
\(\frac{8}{10}=\frac{13}{AB}\)
\(8AB = 130\)
\(AB = \frac{130}{8}\)
\(AB = \frac{65}{4}\)
As the length of AB can be represented as \(\frac{m}{n}\), then \(m+n = 65+4 = 69\)
Answer D
Let the the line created by the two diameters of the two smaller circles = \(xy\), with the two corresponding points being \(x\) and \(y\) respectively.
We know that \(OA = OB = 13\) and that \(xy = 10\). We can work out the length \(Ox\) and \(Oy\) by subtracting the length the radius of the smaller circles from 13. \(13-5 = 8\). Therefore \(Ox\)=\(Oy\)=\(8\).
\(\triangle OAB\) and \(\triangle Oxy\) are similar and therefore we can go: \(\frac{Ox}{xy} = \frac{OA}{OB}\)
\(\frac{8}{10}=\frac{13}{AB}\)
\(8AB = 130\)
\(AB = \frac{130}{8}\)
\(AB = \frac{65}{4}\)
As the length of AB can be represented as \(\frac{m}{n}\), then \(m+n = 65+4 = 69\)
Answer D
Attachments
Circle.png [ 19.09 KiB | Viewed 3114 times ]
27 Sep 2022, 04:22
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Question: The problem mentioned that M and N are prime positive ingetegers. Therefore, M and N cannot be 65 and 4 right?
