2017 AMC 12A Problems/Problem 16
Contents
Problem
In the figure below, semicircles with centers at and
and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter
. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at
is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at
?
Solution 1
Connect the centers of the tangent circles! (call the center of the large circle )
Notice that we don't even need the circles anymore; thus, draw triangle with cevian
:
and use Stewart's Theorem:
From what we learned from the tangent circles, we have ,
,
,
,
, and
, where
is the radius of the circle centered at
that we seek.
Thus:
NOTICE to proficient editors: please label the points on the diagrams, thanks!
Solution 2
Like the solution above, connecting the centers of the circles results in triangle with cevian
. The two triangles
and
share angle
, which means we can use Law of Cosines to set up a system of 2 equations that solve for
respectively:
(notice that the diameter of the largest semicircle is 6, so its radius is 3 and
is 3 - r)
We can eliminate the extra variable of angle by multiplying the first equation by 3 and subtracting the second from it. Then, expand to find
:
, so
=
See Also
2017 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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 12 Problems and Solutions |
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