document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182585>Question 1182585</A>: <I><SPAN STYLE=\"word-break: break-all;\"> It is desired to determine the distance between two <BR>\n" );
document.write( "inaccessible objects A and B. Both A and B are visible <BR>\n" );
document.write( "from two other points C and D. Given the following <BR>\n" );
document.write( "measurements, determine the distance AB: DC = 153.4 m, <BR>\n" );
document.write( "ADB = 58°22’, ADC = 81° 17’, BCA = 65° 43’ and BCD = <BR>\n" );
document.write( "100° 08’ </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849918 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849918\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the distance AB using the given measurements:\r<BR>\n" );
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document.write( "**1. Calculate angles BDC and ABD:**\r<BR>\n" );
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document.write( "* **Angle BDC:**<BR>\n" );
document.write( "   * BDC = BCD - BCA<BR>\n" );
document.write( "   * BDC = 100° 08’ - 65° 43’<BR>\n" );
document.write( "   * BDC = 34° 25’\r<BR>\n" );
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document.write( "* **Angle ABD:**<BR>\n" );
document.write( "   * In triangle ADB, the sum of angles is 180°<BR>\n" );
document.write( "   * ABD = 180° - ADB - BDC<BR>\n" );
document.write( "   * ABD = 180° - 58° 22’ - 34° 25’<BR>\n" );
document.write( "   * ABD = 87° 13’\r<BR>\n" );
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document.write( "**2. Calculate angles ACD and BAC:**\r<BR>\n" );
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document.write( "* **Angle ACD:**<BR>\n" );
document.write( "   * In triangle BCD, the sum of angles is 180°<BR>\n" );
document.write( "   * ACD = 180° - ADC - BCA<BR>\n" );
document.write( "   * ACD = 180° - 81° 17’ - 65° 43’<BR>\n" );
document.write( "   * ACD = 33°\r<BR>\n" );
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document.write( "* **Angle BAC:**<BR>\n" );
document.write( "   * In triangle ABC, the sum of angles is 180°<BR>\n" );
document.write( "   * BAC = 180° - BCA - ACB<BR>\n" );
document.write( "   * We know BCA = 65° 43’ and ACB = BCD - ACD = 100° 08' - 33° = 67° 08'<BR>\n" );
document.write( "   * BAC = 180° - 65° 43’ - 67° 08’<BR>\n" );
document.write( "   * BAC = 47° 09’\r<BR>\n" );
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document.write( "**3. Use the Law of Sines to find AD and AC:**\r<BR>\n" );
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document.write( "* **Side AD:**<BR>\n" );
document.write( "   * AD / sin(BDC) = DC / sin(ADB)<BR>\n" );
document.write( "   * AD = (DC * sin(BDC)) / sin(ADB)<BR>\n" );
document.write( "   * AD = (153.4 * sin(34° 25’)) / sin(58° 22’)<BR>\n" );
document.write( "   * AD ≈ 98.2 m\r<BR>\n" );
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document.write( "* **Side AC:**<BR>\n" );
document.write( "   * AC / sin(BCA) = DC / sin(ADC)<BR>\n" );
document.write( "   * AC = (DC * sin(BCA)) / sin(ADC)<BR>\n" );
document.write( "   * AC = (153.4 * sin(65° 43’)) / sin(81° 17’)<BR>\n" );
document.write( "   * AC ≈ 142.6 m\r<BR>\n" );
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document.write( "**4. Use the Law of Cosines to find AB:**\r<BR>\n" );
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document.write( "* In triangle ABC:<BR>\n" );
document.write( "   * AB² = AC² + BC² - 2 * AC * BC * cos(ACB)<BR>\n" );
document.write( "   * We already know AC and we can find BC using the Law of Sines in triangle BCD.<BR>\n" );
document.write( "   * BC / sin(BDC) = DC / sin(DBC)<BR>\n" );
document.write( "   * We need to find angle DBC = 180 - BDC - BCD = 180 - 34° 25' - 100° 08' = 45° 27'<BR>\n" );
document.write( "   * BC = (153.4 * sin(34° 25')) / sin(45° 27') = 119.8 m<BR>\n" );
document.write( "   * AB² = 142.6² + 119.8² - 2 * 142.6 * 119.8 * cos(67° 08’)<BR>\n" );
document.write( "   * AB ≈ 122.1 m\r<BR>\n" );
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document.write( "**Therefore, the distance between A and B is approximately 122.1 meters.** </SPAN>\n" );
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