document.write( "Question 1079199: Find the equation of the circle that inscribed in the triangle whose sides are 3x - 4y = 19, 4x + 3y = 17 and x + 7 = 0. \n" ); document.write( "

Algebra.Com's Answer #693536 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "Hello,\r\n" );
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document.write( "I don't know what is your level in Math (what is your grade and the level of knowledge).\r\n" );
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document.write( "What I do know that this problem is hard, is of the high level and requires high level technique (virtuoso technique) \r\n" );
document.write( "and high level (adequate) understanding.\r\n" );
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document.write( "So, I will ASSUME that your educational level corresponds to the level of the problem (otherwise why did you start it ?)\r\n" );
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document.write( "Therefore, I will give you only the idea of solution, but will not go into details.\r\n" );
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document.write( "So, first.  Notice that the first and the second lines are perpendicular.\r\n" );
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document.write( "            Also notice that the third line is vertical.\r\n" );
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document.write( "            So, you have right-angled triangle.\r\n" );
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document.write( "Second, you need to find the center of the inscribed circle. \r\n" );
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document.write( "            The center lies at the intersection of the angle bisectors.\r\n" );
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document.write( "            As the first angle bisector, it is naturally to take the angle bisector of the right angle.\r\n" );
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document.write( "            So, find the intersection point of the first two lines (solve the system of 2 equations in two unknowns).\r\n" );
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document.write( "            Next, you know the slopes of these two lines (from their equations).\r\n" );
document.write( "            They are tangents of some canonical angles on the coordinate plane.\r\n" );
document.write( "            By knowing tangents of those angles, find the tangent of the angle bisector of the straight angle. \r\n" );
document.write( "            It will give you the slope of that angle bisector.\r\n" );
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document.write( "            In this way you can obtain the equation of the bisector of the right angle.\r\n" );
document.write( "            As you CAN do it in principle, I will assume that you can do it in reality.\r\n" );
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document.write( "Now, third. You can do the same (or the similar) for the angle bisector of any other acute angle of the triangle.\r\n" );
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document.write( "            In this way you can obtain the equation of the bisector of the second angle.\r\n" );
document.write( "            As you CAN do it in principle, I will assume that you can do it in reality.\r\n" );
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document.write( "Step #4. Having equations of the angle bisectors, you can find their intersection.\r\n" );
document.write( "             It will be the center of the circle.\r\n" );
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document.write( "             By having coordinates of the circle, you can easily complete the solution.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Again, the key idea is: by knowing the slopes of sides, find the slopes of angle bisectors using trigonometric functions (and Trigonometry in general).\r
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document.write( "     What I described above, is not the unique way to solve the problem.\r\n" );
document.write( "     There is another way.\r\n" );
document.write( "     It is to calculate the measures of all three sides;\r\n" );
document.write( "           then to use the property of the angle bisector in any triangle:\r\n" );
document.write( "               \r\n" );
document.write( "                it divides the opposite side proportionally to adjacent sides.\r\n" );
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document.write( "           Using this property, you can find the intersection point for every angle bisector at the opposite side.\r\n" );
document.write( "           Having it, you can construct the equations for each angle bisector.\r\n" );
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\n" ); document.write( "\n" ); document.write( "With this, good luck.\r
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\n" ); document.write( "\n" ); document.write( "As a tutor, I completed my mission.\r
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\n" ); document.write( "\n" ); document.write( "I pointed the way to you.\r
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\n" ); document.write( "\n" ); document.write( "The rest depends on you.\r
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\n" ); document.write( "\n" ); document.write( "Good luck, again !\r
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\n" ); document.write( "\n" ); document.write( "For your info:\r
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\n" ); document.write( "\n" ); document.write( "there are free of charge online textbooks in this site:\r
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\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "\n" ); document.write( "    - GEOMETRY - YOUR ONLINE TEXTBOOK \r
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\n" ); document.write( "\n" ); document.write( "I will be happy (and even more than happy) if you find them useful for you, at least in some parts !!\r
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