Difference between revisions of "2023 AMC 12A Problems/Problem 11"
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+ | We can take any two lines of this form, since the angle between them will always be the same. Let's take <math>y=2x</math> for the line with slope of 2 and <math>y=\frac{1}{3}x</math> for the line with slope of 1/3. Let's take 3 lattice points and create a triangle. Let's use <math>(0,0)</math>, <math>(1,2)</math>, and <math>(3,1)</math>. The distance between the origin and <math>(1,2)</math> is <math>\sqrt{5}</math>. The distance between the origin and <math>(3,1)</math> is <math>\sqrt{10}</math>. The distance between <math>(1,2)</math> and <math>(3,1)</math> is <math>\sqrt{5}</math>. We notice that we have a triangle with 3 side lengths: <math>\sqrt{5}</math>, <math>\sqrt{5}</math>, and <math>\sqrt{10}</math>. This forms a 45-45-90 triangle, meaning that the angle is <math>\boxed{45^\circ}</math>. | ||
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+ | ~lprado | ||
==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}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 19:41, 9 November 2023
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
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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