Question 179604
First, let's find the midpoint of the segment joining (-4,2)and (-2,8)



To find the midpoint, first we need to find the individual coordinates of the midpoint.



<h4>X-Coordinate of the Midpoint:</h4>



To find the x-coordinate of the midpoint, simply average the two x-coordinates of the given points by adding them up and dividing that result by 2 like this:



{{{x[mid]=(-4+-2)/2=-6/2=-3}}}



So the x-coordinate of the midpoint is {{{x=-3}}}



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<h4>Y-Coordinate of the Midpoint:</h4>



To find the y-coordinate of the midpoint, simply average the two y-coordinates of the given points by adding them up and dividing that result by 2 like this:



{{{y[mid]=(2+8)/2=10/2=5}}}



So the y-coordinate of the midpoint is {{{y=5}}}



So the midpoint of the segment joining the points *[Tex \LARGE \left(-4,2\right)] and *[Tex \LARGE \left(-2,8\right)] is *[Tex \LARGE \left(-3,5\right)]



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Now, ANY line parallel to {{{y=-8}}} is a horizontal line of the form {{{y=k}}}. Since the parallel line must pass through the midpoint (-3,5) (and the coordinate of this point is {{{y=5}}}), this means that the parallel line is {{{y=5}}}



Here's a visual confirmation:


{{{ drawing(500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,0,-8,5),
 circle(-4,2,0.05),
 circle(-4,2,0.08),
 circle(-4,2,0.10),
 circle(-2,8,0.05),
 circle(-2,8,0.08),
 circle(-2,8,0.10),
 circle(-3,5,0.05),
 circle(-3,5,0.08),
 circle(-3,5,0.10),
 red(line(-4,2,-2,8))

)}}} 


Graph of {{{y=-8}}} (green) and {{{y=5}}} (blue) through the midpoint (-3,5) of the segment with endpoints (-4,2) and (-2,8)