2022 AIME II Problems/Problem 7
Contents
Problem
A circle with radius is externally tangent to a circle with radius
. Find the area of the triangular region bounded by the three common tangent lines of these two circles.
Solution 1
,
,
,
,
,
,
,
,
Alternative Finish
Note that by AA similarity. Setting up the ratio
and then substituting
with our known values will lead us to the same solution.
mathboy282
Solution 2
Let the center of the circle with radius be labeled
and the center of the circle with radius
be labeled
. Drop perpendiculars on the same side of line
from
and
to each of the tangents at points
and
, respectively. Then, let line
intersect the two diagonal tangents at point
. Since
, we have
Next, throw everything on a coordinate plane with
and
. Then,
, and if
, we have
Combining these and solving, we get
. Notice now that
,
, and the intersections of the lines
(the vertical tangent) with the tangent containing these points are collinear, and thus every slope between a pair of points will have the same slope, which in this case is
. Thus, the other two vertices of the desired triangle are
and
. By the Shoelace Formula, the area of a triangle with coordinates
,
, and
is
~A1001
Solution 3
(Taking diagram names from Solution 1. Also say the line that passes through and is parallel to line EF, call the points of intersection of that line and the circumference of circle
points
and
.)
First notice that is a straight line because
is an isosceles triangle(or you can realize it by symmetry). That means, because
is a straight line, so angle
= angle
triangle
is similar to triangle
. Also name
. By our similar triangles,
. Solving we get
. Pythagorean Theorem on triangle
shows
. By similar triangles,
which means
. Because
.
which means
.
(its value found earlier in this solution) +
(
's radius)
. The area of
is
(because
is
of
)
.
~Professor Rat's solution, added by @heheman and edited by @megahertz13 and @Yrock for .
Solution 4 (similar to solution 1)
First, we want to find . We know that
, so by AA similarity,
. We want to find the length of
, and using the similar triangles, we write an equation:
. Solving, we get
. Therefore,
. Next, we find that using AA similarity,
and they are 3-4-5 triangles. We can quickly compute
. Therefore, the area is
.
~afly
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=7NGkVu0kE08
Video Solution(The Power of Logic)
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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