Difference between revisions of "User:Azjps/1951 AHSME Problems/Problem 3"
m (1951 AHSME Problems/Problem 3 moved to User:Azjps/1951 AHSME Problems/Problem 3: well, this obviously isn't the right question ... move to userspace until we find where it does belong) |
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== Solution == | == Solution == | ||
− | The parabola <math>y=-1/2x^2</math> opens downward, and by symmetry we realize that the y-coordinates of <math>A,B</math> are the same. Thus the segments <math>\overline{AO}, \overline{BO}</math> will have slope <math>\pm \tan{60^{\circ}} = \pm \sqrt{3}</math>. [[Without loss of generality]] consider the equation of <math>AO</math> (we let <math>A</math> be in the third quadrant), which has equation <math>y = \sqrt{3}x</math>. This intersects the graph of <math>y = -\frac{1}{2}x^2</math> at <math>-\frac{1}{2}x^2 = \sqrt{3}x \Longrightarrow x(x + 2\sqrt{3}) = 0</math>; we drop zero so <math>A_x = -2\sqrt{3}</math>. The length of a side of the triangle is <math>|A_x| + |B_x| = 4\sqrt{3}</math> | + | The parabola <math>y=-1/2x^2</math> opens downward, and by symmetry we realize that the y-coordinates of <math>A,B</math> are the same. Thus the segments <math>\overline{AO}, \overline{BO}</math> will have slope <math>\pm \tan{60^{\circ}} = \pm \sqrt{3}</math>. [[Without loss of generality]] consider the equation of <math>AO</math> (we let <math>A</math> be in the third quadrant), which has equation <math>y = \sqrt{3}x</math>. This intersects the graph of <math>y = -\frac{1}{2}x^2</math> at <math>-\frac{1}{2}x^2 = \sqrt{3}x \Longrightarrow x(x + 2\sqrt{3}) = 0</math>; we drop zero so <math>A_x = -2\sqrt{3}</math>. The length of a side of the triangle is <math>|A_x| + |B_x| = 4\sqrt{3}</math>. |
==See Also== | ==See Also== |
Latest revision as of 12:10, 27 August 2020
Problem
Points and are selected on the graph of so that triangle is equilateral. Find the length of one side of triangle (point is at the origin).
Solution
The parabola opens downward, and by symmetry we realize that the y-coordinates of are the same. Thus the segments will have slope . Without loss of generality consider the equation of (we let be in the third quadrant), which has equation . This intersects the graph of at ; we drop zero so . The length of a side of the triangle is .
See Also
1951 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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All AHSME Problems and Solutions |