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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
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+ | https://youtu.be/sa2HHgMZjSg?feature=shared | ||
==See also== | ==See also== | ||
{{AMC12 box|year=2023|ab=A|num-b=10|num-a=12}} | {{AMC12 box|year=2023|ab=A|num-b=10|num-a=12}} | ||
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Revision as of 09:16, 15 December 2023
Contents
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
Solution 4 (Vector Bash)
We can set up vectors and
to represent the two lines. We know that
. Plugging the vectors in gives us
. From this we get that
.
~middletonkids
Solution 5 (Complex Numbers)
Let and
From this we have:
To solve this we must compute
Using elimination we have:
Solution 6
The lines , and
form a large right triangle and a small right triangle. Call the angle that is formed by the x-axis and the line
, and call the angle that is formed by the x-axis and the line
. We try to find
first, and then try to see if any of the answer choices match up.
=
-
.
Using soh-cah-toa, we find that and
.
Plugging it all in, we find that , which is equivalent to
. Since
, we get that
. Therefore, the answer is
.
~Arcticturn
Solution 7 (Cheese)
Using graph paper, as well as the protractor you bought beforehand, draw an accurate to-scale diagram. You can do this by simply drawing the two lines such that they intersect at the origin. Then, measure the angle with a protractor to conclude that they form a 45 degree angle.
The answer is .
~InstallHelp_Hex
Video Solution (Under 4 minutes)
~Education, the Study of Everything
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by Math4All999
https://youtu.be/sa2HHgMZjSg?feature=shared
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.