document.write( "Question 1076741: In right triangle ΔABC (m∠C = 90°), point P is the intersection of the angle bisectors of the acute angles.
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Algebra.Com's Answer #691391 by ikleyn(52908) $\"\"$ $\"About$

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\n" ); document.write( "In right triangle ΔABC (m∠C = 90°), point P is the intersection of the angle bisectors of the acute angles.
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document.write( "The drawing to the problem is shown in the Figure to the right.\r\n" );
document.write( "The angle bisectors are shown in green.\r\n" );
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document.write( "The intersection of angle bisectors is a remarkable point in ANY triangle. NAMELY, \r\n" );
document.write( "it is the center of the inscribed circle and is equidistant from the sides of a triangle.         \r\n" );
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document.write( "    (By the way, the third angle bisector goes through the same intersection point, \r\n" );
document.write( "    since in any triangle the angle bisectors are concurrent.\r\n" );
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document.write( "    See the lesson Angle bisectors of a triangle are concurrent in this site).\r\n" );
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document.write( "Since the condition says that the distance from the point P to the hypotenuse is equal to 4,\r\n" );
document.write( "it means that the radius of the inscribed circle is equal to 4 inches.\r\n" );
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document.write( "The inscribed circle with the center at the point P is shown in red in the Figure.\r\n" );
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document.write( "Also, the radii PQ, PR and PS are drawn (in blue) from the center to the tangent points.\r\n" );
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document.write( " Due to well known property, the radii are perpendicular to the tangent segments, i.e. to the sides of the triangle.\r\n" );
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document.write( "Now, let x be the length of the segment AQ from the vertex A to the tangent point Q, in inches.\r\n" );
document.write( "Then the length of AS is also equal to x inches.\r\n" );
document.write( "Then the length of BS is equal to (20-x) inches.\r\n" );
document.write( "Then the length of BR is equal to (20-x) inches.\r\n" );
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document.write( "It implies that the leg AC of the triangle ABC is (x+4) inches long, while the leg BC is (20-x+4) = (24-x) inches long.\r\n" );
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document.write( "Then we have the Pythagorean equation to find x\r\n" );
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document.write( " = .\r\n" );
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document.write( "After simplifications, it reduces to \r\n" );
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document.write( " = \r\n" );
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document.write( "and is solved by factorization\r\n" );
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document.write( "(x-12)*(x-8) = 0.\r\n" );
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document.write( "The roots are x= 12 and x= 8.\r\n" );
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document.write( "It gives for the legs of the triangle ABC their lengths (16,12) or (12,16).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. The perimeter of the triangle ABC is 16+12+20 = 48 inches.\r
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\n" ); document.write( "\n" ); document.write( "Notice that our triangle ABC is a (3,4,5) right-angled triangle.\r
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\n" ); document.write( "In solution, I used many properties of tangent lines to a circle without direct references.
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\n" ); document.write( "\n" ); document.write( "For those who wants to know more, there is this free of charge online textbook on Geometry\r
\n" ); document.write( "\n" ); document.write( " GEOMETRY - YOUR ONLINE TEXTBOOK \r
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\n" ); document.write( "\n" ); document.write( "See the topic \"Properties of triangles\" (especially properties of angle bisectors in triangles) and\r
\n" ); document.write( "\n" ); document.write( "the topic \"Properties of circles, inscribed angles, chords, secants and tangents.\r
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\n" ); document.write( "\n" ); document.write( " H a p p y l e a r n i n g ! !\r
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