document.write( "Question 1081097: Find the point whose distance from (7, -3) is √58 and whose abscissa equals its ordinate.\r
\n" ); document.write( "\n" ); document.write( "Find the coordinates of the point equidistant from (1, -6), (5, -6) and (6, -1). \n" ); document.write( "
Algebra.Com's Answer #695179 by KMST(5328)\"\" \"About 
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The point \"P%28x%2Cy%29\" whose distance from (7, -3) is \"sqrt%2858%29\"
\n" ); document.write( "is part of the circle with equation
\n" ); document.write( "\"%28x-7%29%5E2%2B%28y%2B3%29%5E2=58\" .
\n" ); document.write( "The point \"P%28x%2Cy%29\" whose abscissa equals its ordinate has \"x=y\" .
\n" ); document.write( "(The equation \"x=y\" is the equation of a line).
\n" ); document.write( "Substituting x for y in the first equation, we get
\n" ); document.write( "\"%28x-7%29%5E2%2B%28x%2B3%29%5E2=58\" .
\n" ); document.write( "We can simplify and solve:
\n" ); document.write( "\"x%5E2-14x%2B49%2Bx%5E2%2B6x%2B9=58\"
\n" ); document.write( "\"2x%5E2-8x%2B58=58\"
\n" ); document.write( "\"2x%5E2-8x=0\"
\n" ); document.write( "\"2x%28x-4%29=0\"
\n" ); document.write( "That last equation's solutions are
\n" ); document.write( "\"x=0\" and \"x=4\" .
\n" ); document.write( "So, the points whose distance from (7, -3) is \"sqrt%2858%29\" ,
\n" ); document.write( "and whose abscissa equals its ordinate are \"highlight%28P%280%2C0%29%29\" and \"highlight%28Q%284%2C4%29%29\" .
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\n" ); document.write( "The points equidistant from \"A%281%2C+-6%29\" and \"B%285%2C+-6%29\" are
\n" ); document.write( "on the perpendicular bisector of AB. and (6, -1).
\n" ); document.write( "The points equidistant from \"A%281%2C+-6%29\" and \"C%286%2C-1%29\" are
\n" ); document.write( "on the perpendicular bisector of AC.
\n" ); document.write( "We can find the equations for those two lines.
\n" ); document.write( "The point equidistant form A, B, and C is in the intersection of those two lines.
\n" ); document.write( "Segment AB is a \"horizontal\" segment (part of the \"y=-6\" line),
\n" ); document.write( "with midpoint \"M%28x%5BM%5D%2Cy%5BM%5D%29\" , with \"x%5BM%5D=%28x%5BA%5D%2Bx%5BB%5D%29%2F2=%281%2B5%29%2F2=3\".
\n" ); document.write( "Its perpendicular bisector is the \"vertical\" line, passing through M,
\n" ); document.write( "and that is the line represented by \"x=3\" .
\n" ); document.write( "Segment AC is part of a line with slope
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\n" ); document.write( "The midpoint of segment AC, \"N%28x%5BN%5D%2Cy%5BN%5D%29\" , has coordinates
\n" ); document.write( "\"x%5BN%5D=%28x%5BA%5D%2Bx%5BC%5D%29%2F2=%281%2B6%29%2F2=7%2F2=3.5\" and
\n" ); document.write( "\"y%5BN%5D=%28y%5BA%5D%2By%5BC%5D%29%2F2=%28-6%2B%28-1%29%29%2F2=%28-7%29%2F2=-7%2F2=-3.5\" .
\n" ); document.write( "The perpendicular bisector of AC has slope \"%28-1%29%2Fm%5BAC%5D=%28-1%29%2F1=-1\" ,
\n" ); document.write( "and passes through \"N%283.5%2C-3.5%29\" , so its equation is
\n" ); document.write( "\"y-%28-3.5%29=%28-1%29%28x-3.5%29\" --> \"y%2B3.5=-x%2B3.5\" --> \"y=-x\" .
\n" ); document.write( "The point equidistant from \"A%281%2C+-6%29\" , \"B%285%2C+-6%29\" and \"C%286%2C-1%29\"
\n" ); document.write( "can be found from
\n" ); document.write( "\"system%28x=3%2Cy=-x%29\" --> \"system%28x=3%2Cy=-3%29\" .
\n" ); document.write( "So, the point equidistant from A, B, and C is \"highlight%28D%283%2C-3%29%29\" .
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\n" ); document.write( "MORE: Let's say you want to find a point \"C%28x%2Cy%29\" that is at a distance \"4sqrt%282%29\" from \"A%28-3%2F2%2C+-5%2F2%29=A%28-1.5%2C-2.5%29\" , and at a distance \"2sqrt%285%29\" from \"B%289%2F2%2C-5%2F2%29=B%284.5%2C-2.5%29\"
\n" ); document.write( "You could draw AB, and use a compass to draw arcs (or circles) centered at A and B,using the given distances as radii, and find the location of point C where the arcs (or circles) intersect.
\n" ); document.write( " You would realize that there are two solutions: \"C\" and its mirror image, \"%22C+%27%22\" .
\n" ); document.write( "You could sketch the triangle ABC; draw the altitude from C to AB, and then apply the Pythagorean theorem to the right triangles formed, to find the location for point C.
\n" ); document.write( "
\n" ); document.write( "You could apply the distance formula.
\n" ); document.write( "When trying to get the coordinates of point C, all those approaches end in the same equations,
\n" ); document.write( "because the distance formula and the equation of a circle are both derived from the Pythagorean theorem.
\n" ); document.write( "The equations you end up with are:
\n" ); document.write( "\"%28x-%28-1.5%29%29%5E2%2B%28y-%28-2.5%29%29%5E2=%284sqrt%282%29%29%5E2\" --> \"%28x%2B1.5%29%5E2%2B%28y%2B2.5%29%5E2=32\"
\n" ); document.write( "\"%28x-4.5%29%5E2%2B%28y-%28-2.5%29%29%5E2=%282sqrt%285%29%29%5E2\" --> \"%28x-4.5%29%5E2%2B%28y%2B2.5%29%5E2=20\"
\n" ); document.write( "Subtracting one equation from the other, we get
\n" ); document.write( "\"%28x%2B1.5%29%5E2-%28x-4.5%29%5E2=32-12\"
\n" ); document.write( "\"%28%28x%2B1.5%29%2B%28x-4.5%29%29%2A%28%28x%2B1.5%29-%28x-4.5%29%29=12\"
\n" ); document.write( "\"%28x%2B1.5%2Bx-4.5%29%2A%28x%2B1.5-x%2B4.5%29=12\"
\n" ); document.write( "\"%282x-3%29%2A6=12\"
\n" ); document.write( "\"2x-3=2\"
\n" ); document.write( "\"2x=5\"
\n" ); document.write( "\"highlight%28x=5%2F2=2.5%29\"
\n" ); document.write( "Substituting that value into either of the original equations, we find to values for \"y\" .
\n" ); document.write( "\"%282.5%2B1.5%29%5E2%2B%28y%2B2.5%29%5E2=32\"
\n" ); document.write( "\"4%5E2%2B%28y%2B2.5%29%5E2=32\"
\n" ); document.write( "\"16%2B%28y%2B2.5%29%5E2=32\"
\n" ); document.write( "\"%28y%2B2.5%29%5E2=16\"
\n" ); document.write( "\"y%2B2.5=%22+%22+%2B-+4\" --> \"y=-2.5+%2B-+4\" --> \"highlight%28system%28y%5B%22C+%27%22%5D=-6.5%2Cy%5BC%5D=1.5%29%29\"
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