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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