2023 AMC 12A Problems/Problem 11
Contents
[hide]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)
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 |
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