document.write( "Question 1088328: Suppose that we have a right triangle $ABC$ with the right angle at $B$ such that $AC = \sqrt{61}$ and $AB = 5.$ A circle is drawn with its center on $AB$ such that the circle is tangent to $AC$ and $BC.$ If $P$ is the point where the circle and side $AC$ meet, then what is $CP$? \n" ); document.write( "

Algebra.Com's Answer #702603 by ikleyn(52803) $\"\"$ $\"About$

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document.write( "0.  Make a sketch to follow my arguments.\r\n" );
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document.write( "    Let O be the center of the circle located on the leg AB, and \r\n" );
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document.write( "        P be the tangent point lying on the hypotenuse AC.\r\n" );
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document.write( "1.  Two right-angled triangles are similar: triangle ABC and triangle APO.\r\n" );
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document.write( "2.  Regarding the triangle ABC, notice that its sides are 5 (the leg AB),  (the hypotenuse AC) and  =  = 6 (the other leg BC).\r\n" );
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document.write( "3.  Let r be the unknown radius of the circle.\r\n" );
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document.write( "4.  Use two proportions that follow the similarity of the triangles:\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "    which after substituting the data takes the form\r\n" );
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document.write( "     = .\r\n" );
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\n" ); document.write( "\n" ); document.write( "From this point, can you complete the solution on your own ?\r
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\n" ); document.write( "If you can - my congratulations !\r
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\n" ); document.write( "\n" ); document.write( "If not, then let me know through the \"thank you\" message/window/form.\r
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\n" ); document.write( "\n" ); document.write( "Then I will help you (at no charge).\r
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\n" ); document.write( "\n" ); document.write( "If you will send a message to me, do not forget to mention the ID number of this problem (# 1088328) in order for I could identify it.\r
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