Difference between revisions of "2018 AMC 10A Problems/Problem 15"
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Latest revision as of 17:10, 15 October 2023
Contents
Problem
Two circles of radius are externally tangent to each other and are internally tangent to a circle of radius
at points
and
, as shown in the diagram. The distance
can be written in the form
, where
and
are relatively prime positive integers. What is
?
Solution
Let the center of the surrounding circle be . The circle that is tangent at point
will have point
as the center. Similarly, the circle that is tangent at point
will have point
as the center. Connect
,
,
, and
. Now observe that
is similar to
by SAS.
Writing out the ratios, we get
Therefore, our answer is
.
Video Solution (HOW TO THINK CREATIVELY!)
~Education, the Study of Everything
Video Solution 1
- Whiz
https://www.youtube.com/watch?v=llMgyOkjNgU&list=PL-27w0UNlunxDTyowGrnvo_T7z92OCvpv&index=3 - amshah
Video Solution 2 by OmegaLearn
https://youtu.be/NsQbhYfGh1Q?t=1328
~ pi_is_3.14
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.