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Difference between revisions of "User:Azjps/1951 AHSME Problems/Problem 3"

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==Problem==
 
==Problem==
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Points <math>A</math> and <math>B</math> are selected on the graph of <math>y=-1/2x^2</math> so that triangle <math>ABO</math> is [[equilateral triangle|equilateral]]. Find the length of one side of triangle <math>ABO</math> (point <math>O</math> is at the origin).
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[[Point]]s <math>A</math> and <math>B</math> are selected on the graph of <math>y=-1/2x^2</math> so that triangle <math>ABO</math> is [[equilateral triangle|equilateral]]. Find the length of one side of triangle <math>ABO</math> (point <math>O</math> is at the origin).
  
 
== Solution ==
 
== Solution ==
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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>. We can now easily verify that this triangle indeed is equilateral.  
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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==
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* [[1951 AHSME]]
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{{AHSME box|year=1951|num-b=2|num-a=4}}
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* [[1952 AHSME Problems/Problem 4 | Next problem]]
 
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]

Latest revision as of 12:10, 27 August 2020

Problem

Points $A$ and $B$ are selected on the graph of $y=-1/2x^2$ so that triangle $ABO$ is equilateral. Find the length of one side of triangle $ABO$ (point $O$ is at the origin).

Solution

The parabola $y=-1/2x^2$ opens downward, and by symmetry we realize that the y-coordinates of $A,B$ are the same. Thus the segments $\overline{AO}, \overline{BO}$ will have slope $\pm \tan{60^{\circ}} = \pm \sqrt{3}$. Without loss of generality consider the equation of $AO$ (we let $A$ be in the third quadrant), which has equation $y = \sqrt{3}x$. This intersects the graph of $y = -\frac{1}{2}x^2$ at $-\frac{1}{2}x^2 = \sqrt{3}x \Longrightarrow x(x + 2\sqrt{3}) = 0$; we drop zero so $A_x = -2\sqrt{3}$. The length of a side of the triangle is $|A_x| + |B_x| = 4\sqrt{3}$.

See Also

1951 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
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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