document.write( "Question 1144732: The coordinates of points A, B, and C are A(-4,9), B(k,0), and C(8,3). What is the value of k, which causes the sum AB+BC to be as small as possible? \n" ); document.write( "

Algebra.Com's Answer #765913 by ikleyn(52787) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            As the problem is formulated, it is not exactly clear what this sum, AB + BC, means.\r
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\n" ); document.write( "\n" ); document.write( "            I will interpret it as the sum of the lengths of segments AB and BC.\r
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\n" ); document.write( "\n" ); document.write( "            My solution is different from the solution by the tutor @greenestamps, and gives another answer.\r
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document.write( "It is a version of the classic Fermat's problem : \r\n" );
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document.write( "    on the plane, a straight line L and two points A and C are given in the same  of the two half-planes.\r\n" );
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document.write( "    Find the shortest path from A to C under the imposed condition, that this path must touch the line L at some point B.\r\n" );
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document.write( "In our case the points are two given points A and C; the straight line L is the x-axis, \r\n" );
document.write( "and we are searching for the point B on this line (on this x-axis)\r\n" );
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document.write( "The solution (very well and widely known) is THIS :  \r\n" );
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document.write( "    - take the point C; reflect it symmetrically about x-axis to get its mirror image C'; \r\n" );
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document.write( "    - then connect the point A with C' by a straight line (which is unique);\r\n" );
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document.write( "    - its intersection with x-axis will be the point B, you are looking for.\r\n" );
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document.write( "So, everything is very easy.\r\n" );
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document.write( "    - The mirror image of the pont C(8,3) is the point C'(8,-3);\r\n" );
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document.write( "    - the straight line connecting A(-4,9) with C'(8,-3) has the slope  m =  =  = -1;\r\n" );
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document.write( "    - the equation of this straight line is  y = 9 - 1*(x-(-4) = 9 - (x+4) = -x + 5;\r\n" );
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document.write( "    - its intersection with the x-aqxis is  at x = 5  (x-intercept);\r\n" );
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document.write( "    - therefore, the point B has coordinates B = (5,0);\r\n" );
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document.write( "    - hence,  k = 5  is the  ANSWER  to the problem's question.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answered, solved, explained in all details and completed.\r
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\n" ); document.write( "\n" ); document.write( "As I presented here this solution for you, it is, usually, one full-scale session of a school Math circle.\r
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\n" ); document.write( "\n" ); document.write( "So, it is as if you visited a first-class school Math circle today at a local University . . . \r
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