First, we need to find the midpoint of the two points in the problem. The formula to find the mid-point of a line segment give the two end points is:
#M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)#
Where #M# is the midpoint and the given points are:
#(color(red)(x_1), color(red)(y_1))# and #(color(blue)(x_2), color(blue)(y_2))#
Substituting gives:
#M = ((color(red)(-8) + color(blue)(-5))/2 , (color(red)(10) + color(blue)(12))/2)#
#M = (-13/2 , 22/2)#
#M = (-6.5, 11)#
Next, we need to find the slope of the line containing the two points in the problem. The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#
Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.
Substituting the values from the points in the problem gives:
#m = (color(red)(12) - color(blue)(10))/(color(red)(-5) - color(blue)(-8)) = (color(red)(12) - color(blue)(10))/(color(red)(-5) + color(blue)(8)) = 2/3#
Now, let's call the slope of the perpendicular line #m_p#. The formula for finding #m_p# is:
#m_p = -1/m#
Substituting gives: #m_p = -1/(2/3) = -3/2#
We can now use the point-slope formula to find an equation for the perpendicular line going through the midpoint of the two points given in the problem. The point-slope form of a linear equation is: #(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))#
Where #(color(blue)(x_1), color(blue)(y_1))# is a point on the line and #color(red)(m)# is the slope.
Substituting the slope we calculated and the values from the mid-point we calculated gives:
#(y - color(blue)(11)) = color(red)(-3/2)(x - color(blue)(-6.5))#
#(y - color(blue)(11)) = color(red)(-3/2)(x + color(blue)(6.5))#
If necessary, we can solve for #y# to put the equation in slope-intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#
Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.
#y - color(blue)(11) = -3/2x + (-3/2 xx color(blue)(6.5))#
#y - color(blue)(11) = -3/2x - 9.75#
#y - color(blue)(11) + 11 = -3/2x - 9.75 + 11#
#y - 0 = -3/2x + 1.25#
#y = color(red)(-3/2)x + color(blue)(1.25)#
