2021 AMC 12B Problems/Problem 11
Problem
Triangle has
and
. Let
be the point on
such that
. There are exactly two points
and
on line
such that quadrilaterals
and
are trapezoids. What is the distance
Solutions
Solution 1 (cheese)
Using Stewart's Theorem of calculate the cevian to be
. It then follows that the answer must also have a factor of the
. Having eliminated 3 answer choices, we then proceed to draw a rudimentary semiaccurate diagram of this figure. Drawing that, we realize that
is too small making out answer
~Lopkiloinm
Solution 2
Using Stewart's Theorem we find . From the similar triangles
and
we have
So
Solution 3
Let be the length
. From the similar triangles
and
we have
Therefore
. Now extend line
to the point
on
, forming parallelogram
. As
we also have
so
.
We now use the Law of Cosines to find (the length of
):
As
, we have (by Law of Cosines on triangle
)
Therefore
And
. The answer is then
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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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