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1961 AHSME Problems/Problem 36 - Revision history
2024-03-28T12:23:16Z
Revision history for this page on the wiki
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Rockmanex3: /* Problem 36 */
2018-06-02T15:15:16Z
<p><span dir="auto"><span class="autocomment">Problem 36</span></span></p>
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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 15:15, 2 June 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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<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>== Problem <del class="diffchange diffchange-inline">36</del>==</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>== Problem ==</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>In <math>\triangle ABC</math> the median from <math>A</math> is given perpendicular to the median from <math>B</math>. If <math>BC=7</math> and <math>AC=6</math>, find the length of <math>AB</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>In <math>\triangle ABC</math> the median from <math>A</math> is given perpendicular to the median from <math>B</math>. If <math>BC=7</math> and <math>AC=6</math>, find the length of <math>AB</math>.  </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7" >Line 7:</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>\textbf{(C)}\ 4.25\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{(C)}\ 4.25\qquad</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{(D)}\ 2\sqrt{5} \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{(D)}\ 2\sqrt{5} \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{(E)}\ 4.5  </math> <del class="diffchange diffchange-inline">  </del></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{(E)}\ 4.5  </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>==Solution==</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>==Solution==</div></td></tr>
</table>
Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1961_AHSME_Problems/Problem_36&diff=94871&oldid=prev
Rockmanex3: Solution to Problem 36
2018-06-02T15:12:51Z
<p>Solution to Problem 36</p>
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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 15:12, 2 June 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</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>\textbf{(E)}\ 4.5  </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)}\ 4.5  </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>==Solution <del class="diffchange diffchange-inline">(WIP)</del>==</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>==Solution==</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><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 colspan="2" class="diff-lineno" id="mw-diff-left-l39" >Line 39:</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;"></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></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="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>By <del class="diffchange diffchange-inline">[[</del>SAS Similarity<del class="diffchange diffchange-inline">]]</del>, <math>\triangle ABC \sim \triangle MNC</math>, so <math>AB \parallel MN</math>.  Thus, by <del class="diffchange diffchange-inline">[[</del>AA Similarity<del class="diffchange diffchange-inline">]]</del>, <math>\triangle AGB \sim \triangle NGM</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>By SAS Similarity, <math>\triangle ABC \sim \triangle MNC</math>, so <math>AB \parallel MN</math>.  Thus, by AA Similarity, <math>\triangle AGB \sim \triangle NGM</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>Let <math>a = GN</math> and <math>b = GM</math>, so <math>AG = 2a</math> and <math>BG = 2b</math>.  By the [[Pythagorean Theorem]],</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>Let <math>a = GN</math> and <math>b = GM</math>, so <math>AG = 2a</math> and <math>BG = 2b</math>.  By the [[Pythagorean Theorem]],</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><cmath>4a^2 + b^2 = 9</cmath></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><cmath>4a^2 + b^2 = 9</cmath></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><cmath>a^2 + 4b^2 = \frac{49}{4}</cmath></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><cmath>a^2 + 4b^2 = \frac{49}{4}</cmath></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;">Adding the two equations yields <math>5a^2 + 5b^2 = \frac{85}{4}</math>, so <math>a^2 + b^2 = \frac{17}{4}</math>.  Thus, <math>MN = \frac{\sqrt{17}}{2}</math>, so <math>AB = \sqrt{17}</math>, which is answer choice <math>\boxed{\textbf{(B)}}</math>.</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>
</table>
Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1961_AHSME_Problems/Problem_36&diff=94870&oldid=prev
Rockmanex3: Created page with "== Problem 36== In <math>\triangle ABC</math> the median from <math>A</math> is given perpendicular to the median from <math>B</math>. If <math>BC=7</math> and <math>AC=6</ma..."
2018-06-02T14:59:54Z
<p>Created page with "== Problem 36== In <math>\triangle ABC</math> the median from <math>A</math> is given perpendicular to the median from <math>B</math>. If <math>BC=7</math> and <math>AC=6</ma..."</p>
<p><b>New page</b></p><div>== Problem 36==<br />
<br />
In <math>\triangle ABC</math> the median from <math>A</math> is given perpendicular to the median from <math>B</math>. If <math>BC=7</math> and <math>AC=6</math>, find the length of <math>AB</math>. <br />
<br />
<math>\textbf{(A)}\ 4\qquad<br />
\textbf{(B)}\ \sqrt{17} \qquad<br />
\textbf{(C)}\ 4.25\qquad<br />
\textbf{(D)}\ 2\sqrt{5} \qquad<br />
\textbf{(E)}\ 4.5 </math> <br />
<br />
==Solution (WIP)==<br />
<asy><br />
<br />
draw((-16,0)--(8,0));<br />
draw((-16,0)--(16,-24));<br />
draw((16,-24)--(0,24)--(0,-12));<br />
draw((-16,0)--(0,24));<br />
draw((0,2)--(2,2)--(2,0));<br />
draw((0,-12)--(8,0),dotted);<br />
<br />
dot((16,-24));<br />
label("C",(16,-24),SE);<br />
dot((-16,0));<br />
label("A",(-16,0),W);<br />
dot((0,24));<br />
label("B",(0,24),N);<br />
<br />
label("3",(8,-18),SW);<br />
label("3",(-8,-6),SW);<br />
label("3.5",(12,-12),NE);<br />
label("3.5",(4,12),NE);<br />
<br />
dot((0,-12));<br />
label("M",(0,-12),SW);<br />
dot((8,0));<br />
label("N",(8,0),NE);<br />
dot((0,0));<br />
label("G",(0,0),NW);<br />
<br />
</asy><br />
By [[SAS Similarity]], <math>\triangle ABC \sim \triangle MNC</math>, so <math>AB \parallel MN</math>. Thus, by [[AA Similarity]], <math>\triangle AGB \sim \triangle NGM</math>.<br />
<br />
Let <math>a = GN</math> and <math>b = GM</math>, so <math>AG = 2a</math> and <math>BG = 2b</math>. By the [[Pythagorean Theorem]],<br />
<cmath>4a^2 + b^2 = 9</cmath><br />
<cmath>a^2 + 4b^2 = \frac{49}{4}</cmath><br />
<br />
==See Also==<br />
{{AHSME 40p box|year=1961|num-b=35|num-a=37}}<br />
<br />
{{MAA Notice}}<br />
<br />
[[Category:Intermediate Geometry Problems]]</div>
Rockmanex3