<PRE><B>Question 1131751</B>
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(corrected/revised response 12/15)&lt;br&gt;
Going from A(0,4) to B(b,0) to C(c,10) to D(8,9), the y values go...
down from 4 to 0;
up from 0 to 10; and
down from 10 to 9&lt;br&gt;
Going from A to B to C to D, the x values go from 0 to b to c to 8.  The minimum total distance from A to B to C to D will be if we are always moving in the same x direction.  Since the starting x value is 0 and the ending x value is 8, we know b and c will be between 0 and 8.&lt;br&gt;
So A to B is a negative slope, B to C is a positive slope, and C to D is a negative slope.&lt;br&gt;
The minimum total length of the path from A to D will be when the three slopes all have the same absolute value.&lt;br&gt;
The slope of AB is {{{-4/b}}}.
The slope of BC is {{{10/(c-b)}}}.
The slope of CD is {{{-1/(8-c)}}}.&lt;br&gt;
So we need to have equal slopes for AB and CD:&lt;br&gt;
{{{-4/b = -1/(8-c)}}}&lt;br&gt;
{{{-b = 4c-32}}}
(1) {{{b = -4c+32}}}&lt;br&gt;
And we need to have the slope of BC the opposite of the slope of CD:&lt;br&gt;
{{{-1/(8-c) = -10/(c-b)}}}&lt;br&gt;
{{{b-c = 10c-80}}}
(2) {{{b = 11c-80}}}&lt;br&gt;
We can eliminate c between (1) and (2) to find the value of b.&lt;br&gt;
(1) {{{11b = -44c+352}}}
(2) {{{4b = 44c-320}}}&lt;br&gt;
{{{15b = 32}}}
{{{b = 32/15}}}&lt;br&gt;
ANSWER:  The minimum distance from A to B to C to D is when b = 32/15.&lt;br&gt;
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(added 12/15)&lt;br&gt;
Perhaps this alternative method of solving the problem makes the solution easier to understand....&lt;br&gt;
We are trying to find the minimum distance from A(0,4) to B(b,0) to C(c,10) to D(8,9).  The first leg from A to B has a negative slope; the second from B to C has a positive slope; the third from C to D has a negative slope.&lt;br&gt;
We can make all the slopes positive as follows:&lt;br&gt;
The distance from (0,4) to (b,0) is the same as the distance from (0,-4) to (b,0).  So define a new point A&#39;(0,-4).&lt;br&gt;
The distance from (c,10) to (8,9) is the same as the distance from (c,10) to (8,11).  So define a new point D&#39;(8,11).&lt;br&gt;
Every leg of the path from A&#39; to B to C to D&#39; now has a positive slope.&lt;br&gt;
The shortest distance from A&#39; to B to C to D&#39; will be the straight line distance from A&#39; to D&#39;.&lt;br&gt;
So to find the value of b, we find the equation of the line A&#39;D&#39; and find the value of b for which (b,0) is on that line.&lt;br&gt;
The slope of line A&#39;D&#39; is (11-(-4))/(8-0) = 15/8; the y-intercept is (0,-4); the equation is&lt;br&gt;
{{{y = (15/8)x-4}}}&lt;br&gt;
Solving for x when y=0:&lt;br&gt;
{{{0 = (15/8)x-4}}}
{{{(15/8)x = 4}}}
{{{x = 32/15}}}</PRE>