Difference between revisions of "1980 AHSME Problems/Problem 12"
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== Solution == | == Solution == | ||
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+ | <math>4n = m</math>, as stated in the question. In the line <math>L_1</math>, draw a triangle with the coordinates <math>(0,0)</math>, <math>(1,0)</math>, and <math>(1,m)</math>. Then <math>m = \tan(\theta_1)</math>. Similarly, <math>n = \tan(\theta_2)</math>. Since <math>4n = m</math> and <math>\theta_1 = 2\theta_2</math>, <math>\tan(2\theta_2) = 4\tan(\theta_2)</math>. Using the angle addition formula for tangents, <math>\dfrac{2\tan(\theta_2)}{1-\tan^2(\theta_2)} = 4\tan(\theta_2)</math>. Solving, we have <math>\tan(\theta_2) = 0, \dfrac{\sqrt{2}}{2}</math>. But line <math>L_1</math> is not horizontal, so therefore <math>(m,n) = (2\sqrt{2},\dfrac{\sqrt{2}}{2})</math>. Looking at the answer choices, it seems the answer is <math>(2\sqrt{2})(\dfrac{\sqrt{2}}{2}) = \boxed{\text{(C)} \ 2}</math>. | ||
== See also == | == See also == | ||
− | {{AHSME | + | {{AHSME box|year=1980|num-b=11|num-a=13}} |
[[Category: Introductory Algebra Problems]] | [[Category: Introductory Algebra Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 18:02, 20 March 2018
Problem
The equations of and are and , respectively. Suppose makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does , and that has 4 times the slope of . If is not horizontal, then is
Solution
Solution by e_power_pi_times_i
, as stated in the question. In the line , draw a triangle with the coordinates , , and . Then . Similarly, . Since and , . Using the angle addition formula for tangents, . Solving, we have . But line is not horizontal, so therefore . Looking at the answer choices, it seems the answer is .
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 |
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