Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

1980 AHSME Problems/Problem 12

Revision as of 15:05, 14 August 2025 by J314andrews (talk | contribs) (Added second solution using Angle Bisector Theorem.)

Problem

The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$, respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis) as does $L_2$, and that $L_1$ has $4$ times the slope of $L_2$. If $L_1$ is not horizontal, then $mn$ is

$\text{(A)} \ \frac{\sqrt{2}}{2} \qquad \text{(B)} \ -\frac{\sqrt{2}}{2} \qquad \text{(C)} \ 2 \qquad \text{(D)} \ -2 \qquad \text{(E)} \ \text{not uniquely determined}$


Solution 1 (Trigonometry)

Since $L_1$ has $4$ times the slope of $L_2$, $m = 4n$. Let $O = (0,0)$, $A = (1,0)$, $B = (1,n)$, $C = (1,m)$, and $\theta = \angle AOB = \angle BOC$. Then $m = \tan 2\theta$ and $n = \tan \theta$.

Since $m = 4n$, $\tan 2\theta = 4\tan \theta$. By the tangent double-angle formula, $\dfrac{2\tan(\theta)}{1-\tan^2(\theta)} = 4\tan(\theta)$. Cross-multiplying and collecting terms on one side yields $4\tan^3 \theta - 2\tan\theta = 0$, which factors as $2\tan \theta(2\tan^2\theta - 1) = 0$. Substituting $\tan\theta = n$ yields $2n(2n^2-1)=0$.

Since line $L_1$ is not horizontal, $n \neq 0$. So $2n^2 - 1 = 0$, and thus $n^2 = \frac{1}{2}$. Therefore, $mn = 4n^2 = 4 \cdot \frac{1}{2} = \boxed{(\textbf{C}) \ 2}$.

-e_power_pi_times_i, edited by j314andrews

Solution 2 (Angle Bisector Theorem)

Since $L_1$ has $4$ times the slope of $L_2$, $m = 4n$. Let $O = (0,0)$, $A = (1,0)$, $B = (1,n)$, $C = (1,m) = (1, 4n)$. By the Angle Bisector Theorem, $\frac{OC}{OA} = \frac{BC}{AB}$. That is, $OC = \frac{3n}{n} = 3$.

By the Pythagorean Theorem, $AC = \sqrt{3^2-1^2} = 2\sqrt{2}$. So $m = 2\sqrt{2}$ and $n = \frac{2\sqrt{2}}{4} = \frac{\sqrt{2}}{2}$. Therefore, $mn = \left(2\sqrt{2}\right)\left(\frac{\sqrt{2}}{2}\right) = \boxed{(\textbf{C}) \ 2}$.

-j314andrews

See also

1980 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1980_AHSME_Problems/Problem_12&oldid=257565"