Difference between revisions of "2023 AMC 12A Problems/Problem 11"
Failure.net (talk | contribs) (→Solution 3 (Law of Cosines)) |
Failure.net (talk | contribs) (→Solution 3 (Law of Cosines)) |
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<math>\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}= \cos(\theta)</math>. | <math>\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}= \cos(\theta)</math>. | ||
− | Thus, <math>\theta = \boxed{45}</math> | + | Thus, <math>\theta = \boxed{\textbf{(C)} 45^\circ}</math> |
~Failure.net | ~Failure.net | ||
Revision as of 00:14, 10 November 2023
Problem
What is the degree measure of the acute angle formed by lines with slopes and
?
Solution 1
Remind that where
is the angle between the slope and
-axis.
,
. The angle formed by the two lines is
.
. Therefore,
.
~plasta
Solution 2
We can take any two lines of this form, since the angle between them will always be the same. Let's take for the line with slope of 2 and
for the line with slope of 1/3. Let's take 3 lattice points and create a triangle. Let's use
,
, and
. The distance between the origin and
is
. The distance between the origin and
is
. The distance between
and
is
. We notice that we have a triangle with 3 side lengths:
,
, and
. This forms a 45-45-90 triangle, meaning that the angle is
.
~lprado
Solution 3 (Law of Cosines)
Follow Solution 2 up until the lattice points section. Let's use ,
, and
. The distance between the origin and
is
. The distance between the origin and
is
. The distance between
and
is
. Using the Law of Cosines, we see the
, where
is the angle we are looking for.
Simplifying, we get .
.
.
.
Thus,
~Failure.net
See also
2023 AMC 12A (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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.