A line segment is bisected by a line with the equation 4 y - 2 x = 5 . If one end of the line segment is at ( 7 , 3 ), where is the other end?

1 Answer
Douglas K.
May 13, 2017

The other end is (28/5,29/5)

Explanation:

Given: 4y-2x=5" [1]" is the equation of the bisector

Find the slope of the bisector:

y = 2/4x+5/4

y = 1/2x+5/4

The slope is m_1= 1/2

The of the bisected line is:

m_2 = -1/m_1

m_2 = -1/(1/2)

m_2=-2

Use the point-slope form of the equation of a line to find the equation of the bisected line:

y = m(x-x_0)+y_0

y = -2(x-7)+3

y = -2x+14+3

y = -2x+17

y + 2x= 17" [2]"

Add equation [2] to equation [1]:

4y+y-2x+2x=5+17

5y = 22

y = 22/5

Use equation [2] to find the corresponding value of x:

22/5+2x=17

2x = 63/5

x = 63/10

Use the midpoint formulas:

x_"midpoint" = (x_"end"+x_"start")/2

y_"midpoint" = (y_"end"+x_"start")/2

Substitute 63/10 for x_"midpoint" and 7 for x_"start":

63/10 = (x_"end"+7)/2

Substitute 22/5 for y_"midpoint" and 3 for y_"start":

22/5 = (y_"end"+3)/2

Solve for x_"end" and y_"end"

63/10 = (x_"end"+7)/2
22/5 = (y_"end"+3)/2

x_"end" = 63/5-7
y_"end"= 44/5-3

x_"end" = 28/5
y_"end"= 29/5

The other end is (28/5,29/5)

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