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Difference between revisions of "1993 AHSME Problems/Problem 25"
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</asy>
 
</asy>
  
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Let <math>S</math> be the set of points on the rays forming the sides of a <math>120^\circ</math> angle, and let <math>P</math> be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles <math>PQR</math> with <math>Q</math> and <math>R</math> in <math>S</math>. (Points <math>Q</math> and <math>R</math> may be on the same ray, and switching the names of <math>Q</math> and <math>R</math> does not create a distinct triangle.) There are
+
Let <math>S</math> be the set of points on the rays forming the sides of a <math>120^{\circ}</math> angle, and let <math>P</math> be a fixed point inside the angle  
 +
on the angle bisector. Consider all distinct equilateral triangles <math>PQR</math> with <math>Q</math> and <math>R</math> in <math>S</math>.  
 +
(Points <math>Q</math> and <math>R</math> may be on the same ray, and switching the names of <math>Q</math> and <math>R</math> does not create a distinct triangle.)  
 +
There are
  
−
<math>\text{(A) exactly 2 such triangles} \quad
+
<math>\text{(A) exactly 2 such triangles} \quad\\
−
\text{(B) exactly 3 such triangles} \quad
+
\text{(B) exactly 3 such triangles} \quad\\
−
\text{(C) exactly 7 such triangles} \quad
+
\text{(C) exactly 7 such triangles} \quad\\
−
\text{(D) exactly 15 such triangles} \quad
+
\text{(D) exactly 15 such triangles} \quad\\
 
\text{(E) more than 15 such triangles} </math>
 
\text{(E) more than 15 such triangles} </math>
  

Revision as of 02:07, 27 September 2014

Problem

[asy] draw((0,0)--(1,sqrt(3)),black+linewidth(.75)); draw((0,0)--(1,-sqrt(3)),black+linewidth(.75)); draw((0,0)--(1,0),dashed+black+linewidth(.75)); dot((1,0)); MP("P",(1,0),E); [/asy]

Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are

$\text{(A) exactly 2 such triangles} \quad\\ \text{(B) exactly 3 such triangles} \quad\\ \text{(C) exactly 7 such triangles} \quad\\ \text{(D) exactly 15 such triangles} \quad\\ \text{(E) more than 15 such triangles}$

Solution

$\fbox{E}$

See also

1993 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Problem 26
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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