https://artofproblemsolving.com/wiki/index.php?title=1960_AHSME_Problems/Problem_15&feed=atom&action=history
1960 AHSME Problems/Problem 15 - Revision history
2024-03-28T23:32:16Z
Revision history for this page on the wiki
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https://artofproblemsolving.com/wiki/index.php?title=1960_AHSME_Problems/Problem_15&diff=94553&oldid=prev
Rockmanex3 at 23:01, 17 May 2018
2018-05-17T23:01:54Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 23:01, 17 May 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l25" >Line 25:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Using the same steps, <math>p=3a</math>, <math>k=\frac{a^2\sqrt{3}}{4}</math>, and <math>r=\frac{a\sqrt{3}}{3}</math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Using the same steps, <math>p=3a</math>, <math>k=\frac{a^2\sqrt{3}}{4}</math>, and <math>r=\frac{a\sqrt{3}}{3}</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Note that <math>P/p = 3A/3a = A/a</math> and <math>R/r = \frac{A\sqrt{3}}{3} \div \frac{a\sqrt{3}}{3} = A/a</math>.  That means <math>P/p = R/r</math>, so the answer is <math>\boxed{\textbf{(B)}}</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Note that <math>P/p = 3A/3a = A/a</math> and <math>R/r = \frac{A\sqrt{3}}{3} \div \frac{a\sqrt{3}}{3} = A/a</math>.  That means <math>P/p = R/r</math>, so the answer is <math>\boxed{\textbf{(B)}}</math><ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{AHSME 40p box|year=1960|num-b=14|num-a=16}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{AHSME 40p box|year=1960|num-b=14|num-a=16}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Introductory Geometry Problems]]</ins></div></td></tr>
</table>
Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1960_AHSME_Problems/Problem_15&diff=94470&oldid=prev
Rockmanex3: /* Solution */
2018-05-14T15:34:13Z
<p><span dir="auto"><span class="autocomment">Solution</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:34, 14 May 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l21" >Line 21:</td>
<td colspan="2" class="diff-lineno">Line 21:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></asy></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></asy></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>First, find <math>P</math>, <math>K</math>, and <math>R</math> in terms of <math>A</math>.  Since all sides of equilateral triangle are the same, <math>P=3A</math>.  From the area formula, <math>K=\frac{A^2\sqrt{3}}{4}</math>.  By using 30-60-90 triangles, <math>R=\frac{A\sqrt{3}}{3}</math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>First, find <math>P</math>, <math>K</math>, and <math>R</math> in terms of <math>A</math>.  Since all sides of <ins class="diffchange diffchange-inline">an [[</ins>equilateral triangle<ins class="diffchange diffchange-inline">]] </ins>are the same, <math>P=3A</math>.  From the area formula, <math>K=\frac{A^2\sqrt{3}}{4}</math>.  By using 30-60-90 triangles, <math>R=\frac{A\sqrt{3}}{3}</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Using the same steps, <math>p=3a</math>, <math>k=\frac{a^2\sqrt{3}}{4}</math>, and <math>r=\frac{a\sqrt{3}}{3}</math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Using the same steps, <math>p=3a</math>, <math>k=\frac{a^2\sqrt{3}}{4}</math>, and <math>r=\frac{a\sqrt{3}}{3}</math>.</div></td></tr>
</table>
Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1960_AHSME_Problems/Problem_15&diff=94443&oldid=prev
Rockmanex3: /* Problem */
2018-05-13T05:20:47Z
<p><span dir="auto"><span class="autocomment">Problem</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 05:20, 13 May 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Line 4:</td>
<td colspan="2" class="diff-lineno">Line 4:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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:  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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:  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \\</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \\</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad <ins class="diffchange diffchange-inline">\\</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}    </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}    </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1960_AHSME_Problems/Problem_15&diff=94395&oldid=prev
Rockmanex3: /* See Also */
2018-05-11T00:14:21Z
<p><span dir="auto"><span class="autocomment">See Also</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 00:14, 11 May 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l28" >Line 28:</td>
<td colspan="2" class="diff-lineno">Line 28:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==See Also==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{AHSME box|year=1960|num-b=14|num-a=16}}</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{AHSME <ins class="diffchange diffchange-inline">40p </ins>box|year=1960|num-b=14|num-a=16}}</div></td></tr>
</table>
Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1960_AHSME_Problems/Problem_15&diff=94378&oldid=prev
Rockmanex3: Solution to Problem 15
2018-05-10T22:41:02Z
<p>Solution to Problem 15</p>
<p><b>New page</b></p><div>==Problem==<br />
<br />
Triangle <math>I</math> is equilateral with side <math>A</math>, perimeter <math>P</math>, area <math>K</math>, and circumradius <math>R</math> (radius of the circumscribed circle). <br />
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: <br />
<br />
<math>\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad<br />
\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \\<br />
\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad<br />
\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad<br />
\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes} </math><br />
<br />
==Solution==<br />
<asy><br />
pair A=(0,50),O=(0,0),B=(43.301,-25),C=(-43.301,-25);<br />
draw(A--B--C--A);<br />
draw(circle(O,50));<br />
draw(B--O--C);<br />
draw(anglemark(C,O,B,200));<br />
draw((0,0)--(0,-25));<br />
label("$60^{\circ}$",(-7,-12));<br />
</asy><br />
<br />
First, find <math>P</math>, <math>K</math>, and <math>R</math> in terms of <math>A</math>. Since all sides of equilateral triangle are the same, <math>P=3A</math>. From the area formula, <math>K=\frac{A^2\sqrt{3}}{4}</math>. By using 30-60-90 triangles, <math>R=\frac{A\sqrt{3}}{3}</math>.<br />
<br />
Using the same steps, <math>p=3a</math>, <math>k=\frac{a^2\sqrt{3}}{4}</math>, and <math>r=\frac{a\sqrt{3}}{3}</math>.<br />
<br />
Note that <math>P/p = 3A/3a = A/a</math> and <math>R/r = \frac{A\sqrt{3}}{3} \div \frac{a\sqrt{3}}{3} = A/a</math>. That means <math>P/p = R/r</math>, so the answer is <math>\boxed{\textbf{(B)}}</math><br />
<br />
==See Also==<br />
{{AHSME box|year=1960|num-b=14|num-a=16}}</div>
Rockmanex3