document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152198>Question 1152198</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let ABCD be an isosceles trapezoid, with bases AB and CD. A circle is inscribed in the trapezoid. (In other words, the circle is tangent to all the sides of the trapezoid.) The length of short base AB is 2x, and the length of long base CD is 2y. Prove that the radius of the inscribed circle is sqrt(xy).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #774160 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=774160\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let E and F be the midpoints of bases AB and CD respectively.  Then AE=x and DF=y; and segment EF is a diameter of the inscribed circle, so its length is 2r, where r is the radius of the inscribed circle.<br><BR>\n" );
document.write( "Let H be on DF such that AH is perpendicular to DF; then the length of AH is also 2r.<br><BR>\n" );
document.write( "Let G be the point of tangency of the inscribed circle with side AD of the trapezoid.<br><BR>\n" );
document.write( "By the theorem about the lengths of tangents to a circle, AG=AE=x and DG=DF=y.<br><BR>\n" );
document.write( "Finally, the length of DH is y-x.<br><BR>\n" );
document.write( "Now look at right triangle ADH.  It has legs of length (y-x) and 2r and hypotenuse of length (x+y).<br><BR>\n" );
document.write( "Use of the Pythagorean Theorem with those lengths will lead you to the desired result.<br><BR>\n" );
document.write( "I leave the calculations to you....<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );