Difference between revisions of "2023 AMC 12A Problems/Problem 11"

Revision as of 00:24, 10 November 2023

Problem

What is the degree measure of the acute angle formed by lines with slopes $2$ and $\frac{1}{3}$ ?

$\textbf{(A)} ~30\qquad\textbf{(B)} ~37.5\qquad\textbf{(C)} ~45\qquad\textbf{(D)} ~52.5\qquad\textbf{(E)} ~60$

Solution 1

Remind that $\text{slope}=\dfrac{\Delta y}{\Delta x}=\tan \theta$ where $\theta$ is the angle between the slope and $x$ -axis. $k_1=2=\tan \alpha$ , $k_2=\dfrac{1}{3}=\tan \beta$ . The angle formed by the two lines is $\alpha-\beta$ . $\tan(\alpha-\beta)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}=\dfrac{2-1/3}{1+2\cdot 1/3}=1$ . Therefore, $\alpha-\beta=\boxed{\textbf{(C)} 45^\circ}$ .

~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 $y=2x$ for the line with slope of 2 and $y=\frac{1}{3}x$ for the line with slope of 1/3. Let's take 3 lattice points and create a triangle. Let's use $(0,0)$ , $(1,2)$ , and $(3,1)$ . The distance between the origin and $(1,2)$ is $\sqrt{5}$ . The distance between the origin and $(3,1)$ is $\sqrt{10}$ . The distance between $(1,2)$ and $(3,1)$ is $\sqrt{5}$ . We notice that we have a triangle with 3 side lengths: $\sqrt{5}$ , $\sqrt{5}$ , and $\sqrt{10}$ . This forms a 45-45-90 triangle, meaning that the angle is $\boxed{45^\circ}$ .

~lprado

Solution 3 (Law of Cosines)

Follow Solution 2 up until the lattice points section. Let's use $(0,0)$ , $(2,4)$ , and $(9,3)$ . The distance between the origin and $(2,4)$ is $\sqrt{20}$ . The distance between the origin and $(9,3)$ is $\sqrt{90}$ . The distance between $(2,4)$ and $(9,3)$ is $\sqrt{50}$ . Using the Law of Cosines, we see the $50 = 90 + 20 - 2\times\sqrt{20}$ $\times\sqrt{90}$ $\times\cos(\theta)$ , where $\theta$ is the angle we are looking for.

Simplifying, we get $-60 = -2\times(\sqrt{20}) \times(\sqrt{90}) \times\cos(\theta)$ .

$30 = \sqrt{1800} \times\cos(\theta)$ .

$30 = 30\sqrt{2} \times\cos(\theta)$ .

$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}= \cos(\theta)$ .

Thus, $\theta = \boxed{\textbf{(C)} 45^\circ}$

~Failure.net

Solution 4 (Vector Bash)

We can set up vectors $\vec{a} = <1,2>$ and $\vec{b} = <3,1>$ to represent the two lines. We know that $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} = \cos \theta$ . Plugging the vectors in gives us $\cos \theta = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}}$ . From this we get that $\theta = \boxed{\textbf{(C)} 45^\circ}$ .

~middletonkids

@@ Line 34: / Line 34: @@
 ==Solution 4 (Vector Bash)==
-We can set up vectors <math>\vec{a} = <1,2></math> and <math>\vec{b} = <3,1></math> to represent the two lines. We know that <math>\frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} = \cos \theta</math>. Plugging this in, we get that <math>\cos \theta = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} </math> From this we get that Thus, <math>\theta = \boxed{\textbf{(C)} 45^\circ}</math>
+We can set up vectors <math>\vec{a} = <1,2></math> and <math>\vec{b} = <3,1></math> to represent the two lines. We know that <math>\frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} = \cos \theta</math>. Plugging the vectors in gives us <math>\cos \theta = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}}</math>.  From this we get that <math>\theta = \boxed{\textbf{(C)} 45^\circ}</math>.
 ~middletonkids

2023 AMC 12A (Problems • Answer Key • Resources)
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