2018 AMC 12A Problems/Problem 23
Problem
In
and
Points
and
lie on sides
and
respectively, so that
Let
and
be the midpoints of segments
and
respectively. What is the degree measure of the acute angle formed by lines
and
Solution
Let be the origin, and
lie on the x axis.
We can find and
Then, we have and
Notice that the tangent of our desired points is the the absolute difference between the y coordinates of the two points divided by the absolute difference between the x coordinates of the two points.
This evaluates to
Now, using sum to product identities, we have this equal to
so the answer is
(lifeisgood03)
Solution 2 (Overkill)
Note that , the midpoint of major arc
on
is the Miquel Point of
(Because
). Then, since
, this spiral similarity carries
to
. Thus, we have
, so
.
But, we have ; thus
.
Then, as is the midpoint of the major arc, it lies on the perpendicular bisector of
, so
. Since we want the acute angle, we have
, so the answer is
.
(stronto)
Solution 3 (Nice, I Think?)
Consider the bisector of This angle makes an
angle with
We claim that
is parallel to this angle bisector, meaning that the answer is
or
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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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