2017 AMC 12A Problems/Problem 16
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[hide]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:
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
=
Solution 3
Let be the center of the largest semicircle and
be the radius of
. We know that
,
,
,
, and
. Notice that
and
are bounded by the same two parallel lines, so these triangles have the same heights. Because the bases of these two triangles (that have the same heights) differ by a factor of 2, the area of
must be twice that of
, since the area of a triangle is
.
Again, we don't need to look at the circle and the semicircles anymore; just focus on the triangles.
Let equal to the area of
and
equal to the area of
.
Heron's Formula states that the area of an triangle with sides
and
is
where
, or the semiperimeter, is
The semiperimeter of
is
Use Heron's Formula to obtain
Using Heron's Formula again, find the area of with sides
,
, and
.
Now,
Solution 4
Let , the center of the large semicircle, to be at
, and
to be at
.
Therefore is at
and
is at
.
Let the radius of circle be
.
Using Distance Formula, we get the following system of three equations:
By simplifying, we get
By subtracting the first equation from the second and third equations, we get
which simplifies to
When we add these two equations, we get
So
solution easiest outline
Draw the other half of the largest circle and proceed with Descartes Circle Theorem.
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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