A line segment is bisected by a line with the equation # 2 y -4 x = 1 #. If one end of the line segment is at #(3 ,8 )#, where is the other end?

1 Answer
Narad T.
Oct 21, 2016

The other end of the line segment is (4.2, 7.4)

Explanation:

Let the other end of the line segment be #(x_1,y_1)#
So the mid-point of #(3,8)# and #(x_1,y_1)# is #((x_1+3)/2,(y_1+8)/2)#
The mid point belongs to the line #2y-4x=1#
The slope of this line is#=2#

Sustituting the values of the mid point in this equation
#(2(y_1+8))/2-4((x_1+3))/2=1#
Simplifying the equation #y_1+8-2(x_1+3)=1#
#y_1+8-2x_1-6=1#
#y_1-2x_1=1-8+6=-1#

We need the equation of the line segment
the slope #m=-1/2# since the two lines are perpendicular and the product of the slopes is #m_1*m_2=-1#

so the equation is #(y_1-8)/(x_1-3)=-1/2#
#2y_1+x_1=19#
#y_1-2x_1=-1#
Solving we get #(y_1=2x_1-1)#
Substituting in other equation
#2(2x_1-1)+x_1=19#
#4x_1+x_1=21# #=>##x_1=21/5=4.2#
and #y_1=-1+2*4.2=7.4#

So the point is (4.2,7.4)

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