document.write( "Question 1147737: A is point (-1,6) on a cartesian graph, and B is point (14,9) on the same graph. Point C is on the x axis. What is the least value of AC+CB? \n" ); document.write( "

Algebra.Com's Answer #769088 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "A is point (-1,6) on a cartesian $\"highlight%28cross%28graph%29%29\"$ $\"highlight%28plane%29\"$ , and B is point (14,9) on the same $\"highlight%28cross%28graph%29%29\"$ $\"highlight%28plane%29\"$ .
\n" ); document.write( "Point C is on the x axis. What is the least value of AC+CB?
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document.write( "It follows the motives of well known (famous) minimization problem, solved about 400 years ago by Pierre Fermat.\r\n" );
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document.write( "The solution is as follows.\r\n" );
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document.write( "(1)  Reflect the point A= (-1,6) about the x-axis as if x-axis is a mirror.\r\n" );
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document.write( "     You will get the point A'= (-1,-6).\r\n" );
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document.write( "(2)  Connect the points A'= (-1,-6)  and  the point B= (14,9) by a straight line.\r\n" );
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document.write( "     \r\n" );
document.write( "     It has the slope m =  =  = 1,\r\n" );
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document.write( "     and its equation is  y - (-6) = 1*(x - (-1)),  which is the same as\r\n" );
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document.write( "     y+6 = x+1,  or  y = x-5.\r\n" );
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document.write( "(3)  Take the x-intercept of this line.\r\n" );
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document.write( "     It is  x= 5, y= 0, i.e. the point  (5,0).\r\n" );
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document.write( "     This point is what the problem asks for :  C = (5,0).\r\n" );
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document.write( "     The distance AC =  =  = .\r\n" );
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document.write( "     The distance BC =  =  = .\r\n" );
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document.write( "ANSWER.  The point C providing minimum sum of lengths  AC + BC  is  C = (5,0).\r\n" );
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document.write( "         The minimum value of  AC + BC  is   +  = .\r\n" );
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