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2005 Alabama ARML TST Problems/Problem 11 - Revision history
2024-03-29T06:29:20Z
Revision history for this page on the wiki
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Xantos C. Guin at 03:31, 17 June 2008
2008-06-17T03:31:02Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 03:31, 17 June 2008</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>==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>
<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>{{<del class="diffchange diffchange-inline">solution</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><ins class="diffchange diffchange-inline">Join four such hexagons and let the points be labeled as shown below: </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> </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 class="diffchange diffchange-inline">[[Image:Fourhexsquare.JPG]]</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> </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 class="diffchange diffchange-inline">Since the sum of the angles in a hexagon is <math>(6-2)180^\circ = 720^\circ</math>, <math>\angle E = 720^\circ - (3\cdot 90^\circ + 80^\circ + 100^\circ) = 270^\circ</math>. </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> </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 class="diffchange diffchange-inline">Since <math>\angle D_iE_iF_i = 360^\circ - \angle E = 360^\circ - 270^\circ = 90^\circ</math> and <math>\angle E_iF_iE_</ins>{<ins class="diffchange diffchange-inline">i+1} = \angle F + \angle D = 80^\circ + 100^\circ = 180^\circ</math>, <math>E_1E_2E_3E_4</math> is a square with a side length of <math>EF+DE = 12</math>. </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> </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 class="diffchange diffchange-inline">Also, Since <math>\angle A_iB_iC_i = \angle B = 90^\circ</math> and <math>\angle B_iA_iB_{i+1} = \angle A + \angle C = 90^\circ + 90^\circ = 180^\circ</math>, <math>B_1B_2B_3B_4</math> is a square with a side length of <math>AB+BC = 7+9 = 16</math>. </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> </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 class="diffchange diffchange-inline">Therefore, <math>4[ABCDEF] + [E_1E_2E_3E_4] = [B_1B_2B_3B_4] \Longrightarrow</math> <math>[ABCDEF] =  \frac{[B_1B_2B_3B_4] - [E_1E_2E_3E_4]}{4} = \frac</ins>{<ins class="diffchange diffchange-inline">16^2-12^2</ins>}<ins class="diffchange diffchange-inline">{4} = \boxed{28</ins>}<ins class="diffchange diffchange-inline"></math>. </ins></div></td></tr>
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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>==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>{{ARML box|year=2005|state=Alabama|num-b=10|num-a=12}}</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>{{ARML box|year=2005|state=Alabama|num-b=10|num-a=12}}</div></td></tr>
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Xantos C. Guin
https://artofproblemsolving.com/wiki/index.php?title=2005_Alabama_ARML_TST_Problems/Problem_11&diff=23873&oldid=prev
1=2: /* See also */
2008-03-04T04:21:46Z
<p><span dir="auto"><span class="autocomment">See also</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 04:21, 4 March 2008</td>
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1=2
https://artofproblemsolving.com/wiki/index.php?title=2005_Alabama_ARML_TST_Problems/Problem_11&diff=23871&oldid=prev
1=2: New page: ==Problem== In concave hexagon <math>ABCDEF</math>, <math>\angle A = \angle B = \angle C = 90^\circ</math>, <math>\angle D = 100^\circ</math>, and <math>\angle F = 80^\circ</math>. Also, ...
2008-03-04T04:21:03Z
<p>New page: ==Problem== In concave hexagon <math>ABCDEF</math>, <math>\angle A = \angle B = \angle C = 90^\circ</math>, <math>\angle D = 100^\circ</math>, and <math>\angle F = 80^\circ</math>. Also, ...</p>
<p><b>New page</b></p><div>==Problem==<br />
In concave hexagon <math>ABCDEF</math>, <math>\angle A = \angle B = \angle C = 90^\circ</math>, <math>\angle D = 100^\circ</math>, and <math>\angle F = 80^\circ</math>. Also, <math>CD=FA</math>, <math>AB=7</math>, <math>BC=9</math>, and <math>EF+DE=12</math>. Compute the area of the hexagon.<br />
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==Solution==<br />
{{solution}}<br />
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==See also==</div>
1=2