document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=8055>Question 8055</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is it possible to calculate the height from a given chord to a given arc, given that the only information available is the length of both the chord and the arc, I have been trying to calculate this for some time but it is in vain?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #4464 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=4464\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  We can solve for the height directly without knowing the radius.<BR>\n" );
document.write( " If the arc length is a.<BR>\n" );
document.write( " Let the radius be r and the angle facing the arc be x (in radians),then <BR>\n" );
document.write( " the arc length a = r x  ...(1)<BR>\n" );
document.write( " Set the chord be c (= AB, M the midpoint of AB, O center of the circle), then<BR>\n" );
document.write( " we have r sin x/2 = c/2...(2) [From at the right triangle AMO]<BR>\n" );
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document.write( " Solve the system (1),(2)(2 equations in two variables)<BR>\n" );
document.write( " for r and angle x.<BR>\n" );
document.write( " Then we obtain the height<BR>\n" );
document.write( " r - r cos(x/2) = r(1 - cos x/2) [since OM = r cos x/2]<BR>\n" );
document.write( " (or by (2) use r cos(x/2) = r sqrt(1- sin^2 (x/2)] = r sqrt(1- c^2/4r^2])<BR>\n" );
document.write( "  Hence, then the height = r(1 - cos x/2) = r(1-sqrt(1- c^2/4r^2]))\r<BR>\n" );
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document.write( " Actually, it is not so easy to solve the system (1),(2). <BR>\n" );
document.write( " Because it involves trig functions , so you may need to use Taylor's<BR>\n" );
document.write( " series or Newtons method to solve for r and x.\r<BR>\n" );
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document.write( " More questions are welcome.\r<BR>\n" );
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document.write( " Kenny </SPAN>\n" );
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