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

Difference between revisions of "1960 AHSME Problems/Problem 15"

m (→‎See Also)
m (→‎Problem)
Line 4: Line 4:
 
Triangle <math>II</math> is equilateral with side <math>a</math>, perimeter <math>p</math>, area <math>k</math>, and circumradius <math>r</math>. If <math>A</math> is different from <math>a</math>, then:  
 
Triangle <math>II</math> is equilateral with side <math>a</math>, perimeter <math>p</math>, area <math>k</math>, and circumradius <math>r</math>. If <math>A</math> is different from <math>a</math>, then:  
  
−
<math>\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad
+
<math>\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad \\
 
\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \\
 
\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \\
−
\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad
+
\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad \\
−
\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad
+
\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad \\
 
\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}    </math>
 
\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}    </math>
  

Revision as of 00:20, 13 May 2018

Problem

Triangle $I$ is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle). Triangle $II$ is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then:

$\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad \\ \textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \\ \textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad \\ \textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad \\ \textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}$

Solution

[asy] pair A=(0,50),O=(0,0),B=(43.301,-25),C=(-43.301,-25); draw(A--B--C--A); draw(circle(O,50)); draw(B--O--C); draw(anglemark(C,O,B,200)); draw((0,0)--(0,-25)); label("$60^{\circ}$",(-7,-12)); [/asy]

First, find $P$, $K$, and $R$ in terms of $A$. Since all sides of equilateral triangle are the same, $P=3A$. From the area formula, $K=\frac{A^2\sqrt{3}}{4}$. By using 30-60-90 triangles, $R=\frac{A\sqrt{3}}{3}$.

Using the same steps, $p=3a$, $k=\frac{a^2\sqrt{3}}{4}$, and $r=\frac{a\sqrt{3}}{3}$.

Note that $P/p = 3A/3a = A/a$ and $R/r = \frac{A\sqrt{3}}{3} \div \frac{a\sqrt{3}}{3} = A/a$. That means $P/p = R/r$, so the answer is $\boxed{\textbf{(B)}}$

See Also

1960 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1960_AHSME_Problems/Problem_15&oldid=94443"